Optimization control method for shock absorber

ABSTRACT

A control system for optimizing the performance of a vehicle suspension system by controlling the damping factor of one or more shock absorbers is described. The control system uses a fitness (performance) function that is based on the physical laws of minimum entropy. The control system uses a fuzzy neural network that is trained by a genetic analyzer. The genetic analyzer uses a fitness function that maximizes information while minimizing entropy production. The fitness function uses a difference between the time differential of entropy from a control signal produced in a learning control module and the time differential of the entropy calculated by a model of the suspension system that uses the control signal as an input The entropy calculation is based on a dynamic model of an equation of motion for the suspension system such that the suspension system is treated as an open dynamic system.

REFERENCE TO RELATED APPLICATION

The present application is a divisional of application Ser. No.09/484,877 filed on Jan. 18, 2000 now U.S. Pat. No. 6,212,466.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to an optimization control method for a shockabsorber having a non-linear kinetic characteristic.

2. Description of the Related Art

Feedback control systems are widely used to maintain the output of adynamic system at a desired value in spite of external disturbanceforces that would move the output away from the desired value. Forexample, a household furnace controlled by a thermostat is an example ofa feedback control system. The thermostat continuously measures the airtemperature of the house, and when the temperature falls below a desiredminimum temperature, the thermostat turns the furnace on. When thefurnace has warmed the air above the desired minimum temperature, thenthe thermostat turns the furnace off. The thermostat-furnace systemmaintains the household temperature at a constant value in spite ofexternal disturbances such as a drop in the outside air temperature.Similar types of feedback control are used in many applications.

A central component in a feedback control system is a controlled object,otherwise known as a process “plant,” whose output variable is to becontrolled. In the above example, the plant is the house, the outputvariable is the air temperature of the house, and the disturbance is theflow of heat through the walls of the house. The plant is controlled bya control system. In the above example, the control system is thethermostat in combination with the furnace. The thermostat-furnacesystem uses simple on-off feedback control to maintain the temperatureof the house. In many control environments, such as motor shaft positionor motor speed control systems, simple on-off feedback control isinsufficient. More advanced control systems rely on combinations ofproportional feedback control, integral feedback control, and derivativefeedback control. Feedback that is the sum of proportional plus integralplus derivative feedback is often referred to as PID control.

The PID control system is a linear control system that is based on adynamic model of the plant. In classical control systems, a lineardynamic model is obtained in the form of dynamic equations, usuallyordinary differential equations. The plant is assumed to be relativelylinear, time invariant, and stable. However, many real-world plants aretime varying, highly nonlinear, and unstable. For example, the dynamicmodel may contain parameters (e.g., masses, inductances, aerodynamiccoefficients, etc.) which are either poorly known or depend on achanging environment. Under these conditions, a linear PID controller isinsufficient.

Evaluating the motion characteristics of a nonlinear plant is oftendifficult, in part due to the lack of a general analysis method.Conventionally, when controlling a plant with nonlinear motioncharacteristics, it is common to find certain equilibrium points of theplant and the motion characteristics of the plant are linearized in avicinity near an equilibrium point. Control is then based on evaluatingthe pseudo (linearized) motion characteristics near the equilibriumpoint. This technique works poorly, if at all, for plants described bymodels that are unstable or dissipative. The optimization control for anon-linear kinetic characteristic of a controlled process has not beenwell developed. A general analysis method for non-linear kineticcharacteristic has not been previously available, so a control devicesuited for the linear-kinetic characteristic is often substituted.Namely, for the controlled process with the non-linear kineticcharacteristic, a suitable balance point for the kinetic characteristicis picked. Then, the kinetic characteristic of the controlled process islinearized in a vicinity of the balance point, whereby the evaluation isconducted relative to pseudo-kinetic characteristics.

However, this method has several disadvantageous. Although theoptimization control may be accurately conducted around the balancepoint, its accuracy decreases beyond this balance point. Further, thismethod cannot typically keep up with various kinds of environmentalchanges around the controlled process.

Shock absorbers used for automobiles and motor cycles are one example ofa controlled process having the non-linear kinetic characteristic. Theoptimization of the non-linear kinetic characteristic has been longsought because vehicle's turning performances and ride are greatlyaffected by the damping characteristic and output of the shockabsorbers.

SUMMARY OF THE INVENTION

The present invention solves these and other problems by providing a newcontrol system for optimizing a shock absorber having a non-linearkinetic characteristic. The new AI control system uses a fitness(performance) function that is based on the physical laws of minimumentropy. This control system can be used to control complex plantsdescribed by nonlinear, unstable, dissipative models. The control systemis configured to use smart simulation techniques for controlling theshock absorber (plant).

In one embodiment, the control system comprises a learning system, suchas a neural network that is trained by a genetic analyzer. The geneticanalyzer uses a fitness function that maximizes sensor information whileminimizing entropy production.

In one embodiment, a suspension control uses a difference between thetime differential (derivative) of entropy from the learning control unitand the time differential of the entropy inside the controlled process(or a model of the controlled process) as a measure of controlperformance. In one embodiment, the entropy calculation is based on athermodynamic model of an equation of motion for a controlled processplant that is treated as an open dynamic system.

The control system is trained by a genetic analyzer. The optimizedcontrol system provides an optimum control signal based on data obtainedfrom one or more sensors. For example, in a suspension system, aplurality of angle and position sensors can be used. In an off-line(laboratory) learning mode, fuzzy rules are evolved using a kineticmodel (or simulation) of a vehicle and is suspension system. Data fromthe kinetic model is provided to an entropy calculator which calculatesinput and output entropy production of the model. The input and outputentropy productions are provided to a fitness function calculator thatcalculates a fitness function as a difference in entropy productionrates for the genetic analyzer. The genetic analyzer uses the fitnessfunction to develop a training signal for the off-line control system.Control parameters from the off-line control system are then provided toan online control system in the vehicle.

In one embodiment, the invention includes a method for controlling anonlinear object (a plant) by obtaining an entropy production differencebetween a time differentiation (dS_(u)/dt) of the entropy of the plantand a time differentiation (dS_(c)/dt) of the entropy provided to theplant from a controller. A genetic algorithm that uses the entropyproduction difference as a fitness (performance) function evolves acontrol rule in an off-line controller. The nonlinear stabilitycharacteristics of the plant are evaluated using a Lyapunov function.The genetic analyzer minimizes entropy and maximizes sensor informationcontent. Control rules from the off-line controller are provided to anonline controller to control suspension system. In one embodiment, theonline controller controls the damping factor of one or more shockabsorbers (dampers) in the vehicle suspension system.

In some embodiments, the control method also includes evolving a controlrule relative to a variable of the controller by means of a geneticalgorithm. The genetic algorithm uses a fitness function based on adifference between a time differentiation of the entropy of the plant(dS_(u)/dt) and a time differentiation (dS_(c)/dt) of the entropyprovided to the plant. The variable can be corrected by using theevolved control rule.

In another embodiment, the invention comprises an AI control systemadapted to control a nonlinear plant. The AI control system includes asimulator configured to use a thermodynamic model of a nonlinearequation of motion for the plant. The thermodynamic model is based on aLyapunov function (V), and the simulator uses the function V to analyzecontrol for a state stability of the plant. The AI control systemcalculates an entropy production difference between a timedifferentiation of the entropy of said plant (dS_(u)/dt) and a timedifferentiation (dS_(c)/dt) of the entropy provided to the plant by alow-level controller that controls the plant. The entropy productiondifference is used by a genetic algorithm to obtain an adaptationfunction in which the entropy production difference is minimized. Thegenetic algorithm provides a teaching signal to a fuzzy logic classifierthat determines a fuzzy rule by using a learning process. The fuzzylogic controller is also configured to form a control rule that sets acontrol variable of the controller in the vehicle.

In yet another embodiment, the invention comprises a new physicalmeasure of control quality based on minimum production entropy and usingthis measure for a fitness function of genetic algorithm in optimalcontrol system design. This method provides a local entropy feedbackloop in the control system. The entropy feedback loop provides foroptimal control structure design by relating stability of the plant(using a Lyapunov function) and controllability of the plant (based onproduction entropy of the control system). The control system isapplicable to all control systems, including, for example, controlsystems for mechanical systems, bio-mechanical systems, robotics,electromechanical systems, etc.

BRIEF DESCRIPTION OF THE DRAWINGS

The advantages and features of the disclosed invention will readily beappreciated by persons skilled in the art from the following detaileddescription when read in conjunction with the drawings listed below.

FIG. 1 is a block diagram illustrating a control system for a shockabsorber.

FIG. 2A is a block diagram showing a fuzzy control unit that estimatesan optimal throttle amount for each shock absorber and outputs signalsaccording to the predetermined fuzzy rule based on the detectionresults.

FIG. 2B is a block diagram showing a learning control unit having afuzzy neural network.

FIG. 3 is a schematic diagram of a four-wheel vehicle suspension systemshowing the parameters of the kinetic models for the vehicle andsuspension system.

FIG. 4 is a detailed view of the parameters associated with theright-front wheel from FIG. 3.

FIG. 5 is a detailed view of the parameters associated with theleft-front wheel from FIG. 3.

FIG. 6 is a detailed view of the parameters associated with theright-rear wheel from FIG. 3.

FIG. 7 is a detailed view of the parameters associated with theleft-rear wheel from

DETAILED DESCRIPTION

FIG. 1 is a block diagram illustrating one embodiment of an optimizationcontrol system 100 for controlling one or more shock absorbers in avehicle suspension system.

This control system 100 is divided in an actual (online) control module102 in the vehicle and a learning (offline) module 101. The learningmodule 101 includes a learning controller 118, such as, for example, afuzzy neural network (FNN). The learning controller (hereinafter “theFNN 118”) can be any type of control system configured to receive atraining input and adapt a control strategy using the training input. Acontrol output from the FNN 118 is provided to a control input of akinetic model 120 and to an input of a first entropy productioncalculator 116. A sensor output from the kinetic model is provided to asensor input of the FNN 118 and to an input of a second entropyproduction calculator 114. An output from the first entropy productioncalculator 116 is provided to a negative input of an adder 119 and anoutput from the second entropy calculator 114 is provided to a positiveinput of the adder 119. An output from the adder 119 is provided to aninput of a fitness (performance) function calculator 112. An output fromthe fitness function calculator 112 is provided to an input of a geneticanalyzer 110. A training output from the genetic analyzer 110 isprovided to a training input of the FNN 118.

The actual control module 102 includes a fuzzy controller 124. Acontrol-rule output from the FNN 118 is provided to a control-rule inputof a fuzzy controller 124. A sensor-data input of the fuzzy controller124 receives sensor data from a suspension system 126. A control outputfrom the fuzzy controller 124 is provided to a control input of thesuspension system 126. A disturbance, such as a road-surface signal, isprovided to a disturbance input of the kinetic model 120 and to thevehicle and suspension system 126.

The actual control module 102 is installed into a vehicle and controlsthe vehicle suspension system 126. The learning module 101 optimizes theactual control module 102 by using the kinetic model 120 of the vehicleand the suspension system 126. After the learning control module 101 isoptimized by using a computer simulation, one or more parameters fromthe FNN 118 are provided to the actual control module 101.

In one embodiment, a damping coefficient control-type shock absorber isemployed, wherein the fuzzy controller 124 outputs signals forcontrolling a throttle in an oil passage in one or more shock absorbersin the suspension system 126.

FIGS. 2A and 2B illustrate one embodiment of a fuzzy controller 200suitable for use in the FNN 118 and/or the fuzzy controller 124. In thefuzzy controller 200, data from one or more sensors is provided to afuzzification interface 204. An output from the fuzzification interface204 is provided to an input of a fuzzy logic module 206. The fuzzy logicmodule 206 obtains control rules from a knowledge-base 202. An outputfrom the fuzzy logic module 206 is provided to a de-fuzzificationinterface 208. A control output from the de-fuzzification interface 208is provided to a controlled process 210 (e.g. the suspension system 126,the kinetic model 120, etc.).

The sensor data shown in FIGS. 1 and 2, can include, for example,vertical positions of the vehicle z₀, pitch angle β, roll angle α,suspension angle η for each wheel, arm angle θ for each wheel,suspension length z₆ for each wheel, and/or deflection Z₁₂ for eachwheel. The fuzzy control unit estimates the optimal throttle amount foreach shock absorber and outputs signals according to the predeterminedfuzzy rule based on the detection results.

The learning module 101 includes a kinetic model 120 of the vehicle andsuspension to be used with the actual control module 101, a learningcontrol module 118 having a fuzzy neural network corresponding to theactual control module 101 (as shown in FIG. 2B), and an optimizer module115 for optimizing the learning control module 118.

The optimizer module 115 computes a difference between a timedifferential of entropy from the FNN 118 (dS_(c)/dt) and a timedifferential of entropy inside the subject process (i.e., vehicle andsuspensions) obtained from the kinetic model 120. The computeddifference is used as a performance function by a genetic optimizer 110.The genetic optimizer 110 optimizes (trains) the FNN 118 by geneticallyevolving a teaching signal. The teaching signal is provided to a fuzzyneural network in the FNN 118. The genetic optimizer 110 optimizes thefuzzy neural network (FNN) such that an output of the FNN, when used asan input to the kinetic module 120, reduces the entropy differencebetween the time differentials of both entropy values.

The fuzzy rules from the FNN 118 are then provided to a fuzzy controller124 in the actual control module 102. Thus, the fuzzy rule (or rules)used in the fuzzy controller 124 (in the actual control module 101), aredetermined based on an output from the FNN 118 (in the learning controlunit), that is optimized by using the kinetic model 120 for the vehicleand suspension.

The genetic algorithm 110 evolves an output signal a based on aperformance function ƒ. Plural candidates for α are produced and thesecandidates are paired according to which plural chromosomes (parents)are produced. The chromosomes are evaluated and sorted from best toworst by using the performance function ƒ. After the evaluation for allparent chromosomes, good offspring chromosomes are selected from amongthe plural parent chromosomes, and some offspring chromosomes arerandomly selected. The selected chromosomes are crossed so as to producethe parent chromosomes for the next generation. Mutation may also beprovided. The second-generation parent chromosomes are also evaluated(sorted) and go through the same evolutionary process to produce thenext-generation (i.e., third-generation) chromosomes. This evolutionaryprocess is continued until it reaches a predetermined generation or theevaluation function ƒ finds a chromosome with a certain value. Theoutputs of the genetic algorithm are the chromosomes of the lastgeneration. These chromosomes become input information a provided to theFNN 118.

In the FNN 118, a fuzzy rule to be used in the fuzzy controller 124 isselected from a set of rules. The selected rule is determined based onthe input information α from the genetic algorithm 110. Using theselected rule, the fuzzy controller 124 generates a control signalC_(dn) for the vehicle and suspension system 126. The control signaladjusts the operation (damping factor) of one or more shock absorbers toproduce a desired ride and handling quality for the vehicle.

The genetic algorithm 110 is a nonlinear optimizer that optimizes theperformance function ƒ. As is the case with most optimizers, the successor failure of the optimization often ultimately depends on the selectionof the performance function ƒ.

The fitness function 112 ƒ for the genetic algorithm 110 is given by

ƒ=min dS/dt  (1)

where $\begin{matrix}{\frac{S}{t} = \left( {\frac{S_{c}}{t} - \frac{S_{u}}{t}} \right)} & (2)\end{matrix}$

The quantity dS_(u)/dt represents the rate of entropy production in theoutput x(t) of the kinetic model 120. The quantity dS_(c)/dt representsthe rate of entropy production in the output C_(dn) of the FNN 118.

Entropy is a concept that originated in physics to characterize theheat, or disorder, of a system. It can also be used to provide a measureof the uncertainty of a collection of events, or, for a random variable,a distribution of probabilities. The entropy function provides a measureof the lack of information in the probability distribution. Toillustrate, assume that p(x) represents a probabilistic description ofthe known state of a parameter, that p(x) is the probability that theparameter is equal to z. If p(x) is uniform, then the parameter p isequally likely to hold any value, and an observer will know little aboutthe parameter p. In this case, the entropy function is at its maximum.However, if one of the elements of p(z) occurs with a probability ofone, then an observer will know the parameter p exactly and havecomplete information about p. In this case, the entropy of p(x) is atits minimum possible value. Thus, by providing a measure of uniformity,the entropy function allows quantification of the information on aprobability distribution.

It is possible to apply these entropy concepts to parameter recovery bymaximizing the entropy measure of a distribution of probabilities whileconstraining the probabilities so that they satisfy a statistical modelgiven measured moments or data. Though this optimization, thedistribution that has the least possible information that is consistentwith the data may be found. In a sense, one is translating all of theinformation in the data into the form of a probability distribution.Thus, the resultant probability distribution contains only theinformation in the data without imposing additional structure. Ingeneral, entropy techniques are used to formulate the parameters to berecovered in terms of probability distributions and to describe the dataas constraints for the optimization. Using entropy formulations, it ispossible to perform a wide range of estimations, address ill-posedproblems, and combine information from varied sources without having toimpose strong distributional assumptions.

Entropy-based optimization of the FNN is based on obtaining thedifference between a time differentiation (dS_(u)/dt) of the entropy ofthe plant and a time differentiation (dS_(c)/dt) of the entropy providedto the kinetic model from the FNN 118 controller that controls thekinetic model 120, and then evolving a control rule using a geneticalgorithm. The time derivative of the entropy is called the entropyproduction rate. The genetic algorithm 110 minimizes the differencebetween the entropy production rate of the kinetic model 120 (that is,the entropy production of the controlled process) (dS_(u)/dt) and theentropy production rate of the low-level controller (dS_(c)/dt) as aperformance function. Nonlinear operation characteristics of the kineticmodel (the kinetic model represents a physical plant) are calculated byusing a Lyapunov function.

The dynamic stability properties of the model 120 near an equilibriumpoint can be determined by use of Lyapunov functions as follows. LetV(x) be a continuously differentiable scalar function defined in adomain D⊂R^(n) that contains the origin. The function V(x) is said to bepositive definite if V(0)=0 and V(x)>0 for x≠0. The function V(x) issaid to be positive semidefinite if V(x)≧0 for all x. A function V(x) issaid to be negative definite or negative semidefinite if −V(x) ispositive definite or positive semidefinite, respectively. The derivativeof V along the trajectories {dot over (x)}=ƒ(x) is given by:$\begin{matrix}{{\overset{.}{V}(x)} = {{\sum\limits_{i = 1}^{n}{\frac{\partial V}{\partial x_{i}}{\overset{.}{x}}_{i}}} = {\frac{\partial V}{\partial x}{f(x)}}}} & (3)\end{matrix}$

where ∂V/∂x is a row vector whose ith component is ∂V/∂x_(i) and thecomponents of the n-dimensional vector ƒ(x) are locally Lipschitzfunctions of x, defined for all x in the domain D. The Lyapunovstability theorem states that the origin is stable if there is acontinuously differentiable positive definite function V(x) so that {dotover (V)}(x) is negative definite. A function V(x) satisfying theconditions for stability is called a Lyapunov function.

The genetic algorithm realizes 110 the search of optimal controllerswith a simple structure using the principle of minimum entropyproduction.

The fuzzy tuning rules are shaped by the learning system in the fuzzyneural network 118 with acceleration of fuzzy rules on the basis ofglobal inputs provided by the genetic algorithm 110.

In general, the equation of motion for non-linear systems is expressedas follows by defining “q” as generalized coordinates, “f” and “g”random functions, “Fe” as control input.

q=ƒ({dot over (q)},q)+g(q)−F_(e)  (a)

In the above equation, when the dissipation term and control input inthe second term are multiplied by a speed, the following equation can beobtained for the time differentials of the entropy. $\begin{matrix}{\frac{S}{t} = {{{{f\left( {\overset{.}{q},q} \right)}\overset{.}{q}} - {Feq}} = {\frac{S_{u}}{t} - \frac{S_{c}}{t}}}} & (b)\end{matrix}$

dS/dt is a time differential of entropy for the entire system. dS_(u)/dtis a time differential of entropy for the plant, that is the controlledprocess. dS_(c)/dt is a time differential of entropy for the controlsystem for the plant.

The following equation is selected as Lyapunov function for the equation(a).

V=(Σq²+S²)/2=(Σq²+(S_(u)−S_(c))²)/2  (c)

The greater the integral of the Lyapunov function, the more stable thekinetic characteristic of the plant.

Thus, for the stabilization of the systems, the following equation isintroduced as a relationship between the Lyapunov function and entropyproduction for the open dynamic system.

DV/dt=Σqq+(S_(u)−S_(c))(dS_(u)/dt−ds_(c)/dt)<0  (d)

Σqq<(S_(u)−S_(c))(dS_(c)/dt−ds_(u)/dt)  (e)

A Duffing oscillator is one example of a dynamic system. In the Duffingoscillator, the equation of motion is expressed as:

{umlaut over (x)}+{dot over (x)}+x+x³=0  (f)

the entropy production from this equation is calculated as:

dS/dt=x³  (g)

Further, Lyapunov function relative to the equation (f) becomes:

V=(½)x²+U(x), U(x)=(¼)x⁴−(½)x²  (h)

If the equation (f) is modified by using the equation (h), it isexpressed as: $\begin{matrix}{{\overset{¨}{x} + x + \frac{\partial{U(x)}}{\partial x}} = 0} & (i)\end{matrix}$

If the left side of the equation (i) is multiplied by x as:${\overset{¨}{x} + x + {\frac{\partial{U(x)}}{\partial x}x}} = 0$

Then, if the Lyapunov function is differentiated by time, it becomes:

dV/dt=xx+(∂U(x)/∂x)x

If this is converted to a simple algebra, it becomes:

dV/dt=(1/T)(dS/dt)  (j)

wherein “T” is a normalized factor.

dS/dt is used for evaluating the stability of the system. dS_(u)/dt is atime change of the entropy for the plant. −dS_(c)/dt is considered to bea time change of negative entropy given to the plant from the controlsystem.

The present invention calculates waste such as disturbances for theentire control system of the plant based on a difference between thetime differential dS_(u)/dt of the entropy of the plant that is acontrolled process and time differential dS_(u)/dt of the entropy of theplant. Then, the evaluation is conducted by relating to the stability ofthe controlled process that is expressed by Lyapunov function. In otherwords, the smaller the difference of both entropy, the more stable theoperation of the plants.

Suspension Control

In one embodiment, the control system 100 of FIGS. 1-2 is applied to asuspension control system, such as, for example, in an automobile,truck, tank, motorcycle, etc.

FIG. 3 is a schematic diagram of an automobile suspension system. InFIG. 3, a right front wheel 301 is connected to a right arm 313. Aspring and damper linkage 334 controls the angle of the arm 313 withrespect to a body 310. A left front wheel 302 is connected to a left arm323 and a spring and damper 324 controls the angle of the arm 323. Afront stabilizer 330 controls the angle of the left arm 313 with respectto the right arm 323. Detail views of the four wheels are shown in FIGS.4-7. Similar linkages are shown for a right rear wheel 303 and a leftrear wheel 304.

In one embodiment of the suspension control system, the learning module101 uses a kinetic model 120 for the vehicle and suspension. FIG. 3illustrates each parameter of the kinetic models for the vehicle andsuspensions. FIGS. 4-7 illustrate exploded views for each wheel asillustrated in FIG. 3.

A kinetic model 120 for the suspension system in the vehicle 300 shownin FIGS. 3-7 is developed as follows.

1. Description of transformation matrices

1.1 A Global reference coordinate x_(r), y_(r), z_(r){r} is assumed tobe at the geometric center P_(r) of the vehicle body 310.

The following are the transformation matrices to describe the localcoordinates for:

{2} is a local coordinate in which an origin is the center of gravity ofthe vehicle body 310;

{7} is a local coordinate in which an origin is the center of gravity ofthe suspension;

{10 n} is a local coordinate in which an origin is the center of gravityof the n'th arm;

{12 n} is a local coordinate in which an origin is the center of gravityof the n'th wheel;

{13 n} is a local coordinate in which an origin is a contact point ofthe n'th wheel relative to the road surface; and

{14} is a local coordinate in which an origin is a connection point ofthe stabilizer.

Note that in the development that follows, the wheels 302, 301, 304, and303 are indexed using “i”, “ii”, “iii”, and “iv”, respectively.

1.2 Transformation matrices.

As indicated, “n” is a coefficient indicating wheel positions such as i,ii, iii, and iv for left front, right front, left rear and right rearrespectively. The local coordinate systems x₀, y₀, and z₀{0} areexpressed by using the following conversion matrix that moves thecoordinate {r} along a vector (0,0,z₀) ${\,_{0}^{r}T} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & z_{0} \\0 & 0 & 0 & 1\end{bmatrix}$

Rotating the vector {r} along y_(r) with an angle β makes a localcoordinate system x_(0c), y_(0c), z_(0c){0 r} with a transformationmatrix _(0c) ⁰T. $\begin{matrix}{{\,_{0c}^{0}T} = \begin{bmatrix}{\cos \quad \beta} & 0 & {\sin \quad \beta} & 0 \\0 & 1 & 0 & 0 \\{{- \sin}\quad \beta} & 0 & {\cos \quad \beta} & 0 \\0 & 0 & 0 & 1\end{bmatrix}} & (4)\end{matrix}$

Transferring {0 r} through the vector (a_(1n), 0, 0) makes a localcoordinate system x_(0f), y_(0f), z_(0f){0 f} with a transformationmatrix ^(0r) _(0f)T. $\begin{matrix}{{\,_{0n}^{0c}T} = \begin{bmatrix}1 & 0 & 0 & a_{1n} \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}} & (5)\end{matrix}$

The above procedure is repeated to create other local coordinate systemswith the following transformation matrices. $\begin{matrix}{{\,_{1n}^{0n}T} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \alpha} & {{- \sin}\quad \alpha} & 0 \\0 & {\sin \quad \alpha} & {\cos \quad \alpha} & 0 \\0 & 0 & 0 & 1\end{bmatrix}} & (6) \\{{\,_{2}^{1i}T} = \begin{bmatrix}1 & 0 & 0 & a_{0} \\0 & 1 & 0 & b_{0} \\0 & 0 & 1 & c_{0} \\0 & 0 & 0 & 1\end{bmatrix}} & (7)\end{matrix}$

1.3 Coordinates for the wheels (index n: i for the left front, ii forthe right front, etc.) are generated as follows.

Transferring {1 n} through the vector (0, b_(2n), 0) makes localcoordinate system x_(3n), y_(3n), z_(3n), {3 n} with transformationmatrix ^(1f) _(3n)T. $\begin{matrix}{{\,_{3n}^{1n}T} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & b_{2n} \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}} & (8) \\{{\,_{4n}^{3n}T} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \gamma_{n}} & {{- \sin}\quad \gamma_{n}} & 0 \\0 & {\sin \quad \gamma_{n}} & {\cos \quad \gamma_{n}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}} & (9) \\{{\,_{5n}^{4n}T} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & c_{1n} \\0 & 0 & 0 & 1\end{bmatrix}} & (10) \\{{\,_{6n}^{5n}T} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \eta_{n}} & {{- \sin}\quad \eta_{n}} & 0 \\0 & {\sin \quad \eta_{n}} & {\cos \quad \eta_{n}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}} & (11) \\{{\,_{7n}^{6n}T} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & z_{6n} \\0 & 0 & 0 & 1\end{bmatrix}} & (12) \\{{\,_{8n}^{4n}T} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & c_{2n} \\0 & 0 & 0 & 1\end{bmatrix}} & (13) \\{{\,_{9n}^{8n}T} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \theta_{n}} & {{- \sin}\quad \theta_{n}} & 0 \\0 & {\sin \quad \theta_{n}} & {\cos \quad \theta_{n}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}} & (14) \\{{\,_{10n}^{9n}T} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & e_{1n} \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}} & (15) \\{{\,_{11n}^{9n}T} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & e_{3n} \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}} & (16) \\{{\,_{12n}^{11n}T} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \zeta_{n}} & {{- \sin}\quad \zeta_{n}} & 0 \\0 & {\sin \quad \zeta_{n}} & {\cos \quad \zeta_{n}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}} & (17) \\{{\,_{13n}^{12n}T} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & z_{12n} \\0 & 0 & 0 & 1\end{bmatrix}} & (18) \\{{\,_{14n}^{9n}T} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & e_{0n} \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}} & (19)\end{matrix}$

1.4 Some matrices are sub-assembled to make the calculation simpler.$\begin{matrix}\begin{matrix}{{\,_{1n}^{r}T} = \quad {{\,_{0}^{r}T}{\,_{0n}^{0c}T}{\,_{1n}^{0n}T}}} \\{= \quad {{\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & z_{0} \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}{\cos \quad \beta} & 0 & {\sin \quad \beta} & 0 \\0 & 1 & 0 & 0 \\{{- \sin}\quad \beta} & 0 & {\cos \quad \beta} & 0 \\0 & 0 & 0 & 1\end{bmatrix}}\begin{bmatrix}1 & 0 & 0 & a_{1n} \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}}} \\{\quad \begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \alpha} & {{- \sin}\quad \alpha} & 0 \\0 & {\sin \quad \alpha} & {\cos \quad \alpha} & 0 \\0 & 0 & 0 & 1\end{bmatrix}} \\{= \quad {\begin{bmatrix}{\cos \quad \beta} & 0 & {\sin \quad \beta} & {a_{1n}\cos \quad \beta} \\0 & 1 & 0 & 0 \\{{- \sin}\quad \beta} & 0 & {\cos \quad \beta} & {z_{0} - {a_{1}\sin \quad \beta}} \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \alpha} & {{- \sin}\quad \alpha} & 0 \\0 & {\sin \quad \alpha} & {\cos \quad \alpha} & 0 \\0 & 0 & 0 & 1\end{bmatrix}}} \\{= \quad \begin{bmatrix}{\cos \quad \beta} & {\sin \quad {\beta sin}\quad \alpha} & {\sin \quad {\beta cos\alpha}} & {a_{1n}\cos \quad \beta} \\0 & {\cos \quad \alpha} & {{- \sin}\quad \alpha} & 0 \\{{- \sin}\quad \beta} & {\cos \quad {\beta sin}\quad \alpha} & {\cos \quad {\beta cos}\quad \alpha} & {z_{0} - {a_{1n}\sin \quad \beta}} \\0 & 0 & 0 & 1\end{bmatrix}}\end{matrix} & (20) \\\begin{matrix}{{\,_{4n}^{r}T} = \quad {{\,_{1n}^{r}T}{\,_{3n}^{1n}T}{\,_{4n}^{3n}T}}} \\{= \quad \begin{bmatrix}{\cos \quad \beta} & {\sin \quad {\beta sin}\quad \alpha} & {\sin \quad \beta \quad \cos \quad \alpha} & {a_{1n}\cos \quad \beta} \\0 & {\cos \quad \alpha} & {{- \sin}\quad \alpha} & 0 \\{{- \sin}\quad \beta} & {\cos \quad {\beta sin}\quad \alpha} & {\cos \quad {\beta cos}\quad \alpha} & {z_{0} - {a_{1n}\sin \quad \beta}} \\0 & 0 & 0 & 1\end{bmatrix}} \\{\quad {\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & b_{2n} \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix} \cdot \begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \gamma_{n}} & {{- \sin}\quad \gamma_{n}} & 0 \\0 & {\sin \quad \gamma_{n}} & {\cos \quad \gamma_{n}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}}} \\{= \quad \begin{bmatrix}{\cos \quad \beta} & {\sin \quad {{\beta sin}\left( {\alpha + \gamma_{n}} \right)}} & {\sin \quad {{\beta cos}\left( {\alpha + \gamma_{n}} \right)}} & {{b_{2n}\sin \quad {\beta sin}\quad \alpha} + {a_{1n}\cos \quad \beta}} \\0 & {\cos \left( {\alpha + \gamma_{n}} \right)} & {- {\sin \left( {\alpha + \gamma_{n}} \right)}} & {b_{2n}\cos \quad \alpha} \\{{- \sin}\quad \beta} & {\cos \quad {{\beta sin}\left( {\alpha + \gamma_{n}} \right)}} & {\cos \quad {{\beta cos}\left( {\alpha + \gamma_{n}} \right)}} & {z_{0} - {b_{2n}\cos \quad {\beta sin}\quad \alpha} - {a_{1n}\sin \quad \beta}} \\0 & 0 & 0 & 1\end{bmatrix}}\end{matrix} & (21) \\\begin{matrix}{{\,_{7n}^{4n}T} = \quad {{\,_{5n}^{4n}T}{\,_{6n}^{5n}T}{\,_{7n}^{6n}T}}} \\{= \quad {{\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & c_{1n} \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \eta_{n}} & {{- \sin}\quad \eta_{n}} & 0 \\0 & {\sin \quad \eta_{n}} & {\cos \quad \eta_{n}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & z_{6n} \\0 & 0 & 0 & 1\end{bmatrix}}} \\{= \quad {\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \eta_{n}} & {{- \sin}\quad \eta_{n}} & 0 \\0 & {\sin \quad \eta_{n}} & {\cos \quad \eta_{n}} & c_{1n} \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & z_{6n} \\0 & 0 & 0 & 1\end{bmatrix}}} \\{= \quad \begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \eta_{n}} & {{- \sin}\quad \eta_{n}} & {{- z_{6n}}\sin \quad \eta_{n}} \\0 & {\sin \quad \eta_{n}} & {\cos \quad \eta_{n}} & {c_{1n} + {z_{6n}\cos \quad \eta_{n}}} \\0 & 0 & 0 & 1\end{bmatrix}}\end{matrix} & (22) \\\begin{matrix}{{\,_{10n}^{4n}T} = \quad {{\,_{8n}^{4n}T}{\,_{9n}^{8n}T}{\,_{10n}^{9n}T}}} \\{= \quad {{\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & c_{2n} \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \theta_{n}} & {{- \sin}\quad \theta_{n}} & 0 \\0 & {\sin \quad \theta_{n}} & {\cos \quad \theta_{n}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & e_{1n} \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}}} \\{= \quad {\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \theta_{n}} & {{- \sin}\quad \theta_{n}} & 0 \\0 & {\sin \quad \theta_{n}} & {\cos \quad \theta_{n}} & c_{2n} \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & e_{1n} \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}}} \\{= \quad \begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \theta_{n}} & {{- \sin}\quad \theta_{n}} & {e_{1n}\cos \quad \theta_{n}} \\0 & {\sin \quad \theta_{n}} & {\cos \quad \theta_{n}} & {c_{2n} + {e_{1n}\sin \quad \theta_{n}}} \\0 & 0 & 0 & 1\end{bmatrix}}\end{matrix} & (23) \\\begin{matrix}{{\,_{12n}^{4n}T} = \quad {{\,_{8n}^{4n}T}{\,_{9n}^{8n}T}{\,_{11n}^{9n}T}{\,_{12n}^{11n}T}}} \\{= \quad {{\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & c_{2n} \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \theta_{n}} & {{- \sin}\quad \theta_{n}} & 0 \\0 & {\sin \quad \theta_{n}} & {\cos \quad \theta_{n}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & e_{3n} \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}}} \\{\quad \begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \zeta_{n}} & {{- \sin}\quad \zeta_{n}} & 0 \\0 & {\sin \quad \zeta_{n}} & {\cos \quad \zeta_{n}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}} \\{= \quad {{\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \theta_{n}} & {{- \sin}\quad \theta_{n}} & 0 \\0 & {\sin \quad \theta_{n}} & {\cos \quad \theta_{n}} & c_{2n} \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & e_{3n} \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \zeta_{n}} & {{- \sin}\quad \zeta_{n}} & 0 \\0 & {\sin \quad \zeta_{n}} & {\cos \quad \zeta_{n}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}}} \\{= \quad {\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \theta_{n}} & {{- \sin}\quad \theta_{n}} & {e_{3n}\cos \quad \theta_{n}} \\0 & {\sin \quad \theta_{n}} & {\cos \quad \theta_{n}} & {c_{2n} + {e_{3n}\sin \quad \theta_{n}}} \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \zeta_{n}} & {{- \sin}\quad \zeta_{n}} & 0 \\0 & {\sin \quad \zeta_{n}} & {\cos \quad \zeta_{n}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}}} \\{= \quad \begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \left( {\theta_{n} + \zeta_{n}} \right)} & {- {\sin \left( {\theta_{n} + \zeta_{n}} \right)}} & {e_{3n}\cos \quad \theta_{n}} \\0 & {\sin \quad \left( {\theta_{n} + \zeta_{n}} \right)} & {\cos \left( {\theta_{n} + \zeta_{n}} \right)} & {c_{2n} + {e_{3n}\sin \quad \theta_{n}}} \\0 & 0 & 0 & 1\end{bmatrix}}\end{matrix} & (24)\end{matrix}$

2. Description of all the parts of the model both in local coordinatesystems and relation to the coordinate {r} or {1 n} referenced to thevehicle body 310.

2.1 Description in local coordinate systems. $\begin{matrix}{P_{body}^{2} = {P_{{susp}.n}^{7n} = {P_{{arm}.n}^{10n} = {P_{{wheel}.n}^{12n} = {P_{{touchpoint}.n}^{13n} = {P_{{stab}.n}^{14n} = \begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}}}}}}} & (25)\end{matrix}$

2.2 Description in global reference coordinate system {r}.$\begin{matrix}\begin{matrix}{P_{body}^{r} = \quad {{\,_{1i}^{r}T}{{}_{}^{}{}_{}^{}}}} \\{= \quad \begin{bmatrix}{\cos \quad \beta} & {\sin \quad {\beta sin}\quad \alpha} & {\sin \quad {\beta cos}\quad \alpha} & {a_{1i}\cos \quad \beta} \\0 & {\cos \quad \alpha} & {{- \sin}\quad \alpha} & 0 \\{{- \sin}\quad \beta} & {\cos \quad {\beta sin}\quad \alpha} & {\cos \quad \beta \quad \cos \quad \alpha} & {z_{0} - {a_{1i}\sin \quad \beta}} \\0 & 0 & 0 & 1\end{bmatrix}} \\{\quad {\begin{bmatrix}1 & 0 & 0 & a_{0} \\0 & 1 & 0 & b_{0} \\0 & 0 & 1 & c_{0} \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}}} \\{= \quad \begin{bmatrix}{{{a_{0}\cos \quad \beta} + {b_{0}\sin \quad {\beta sin}\quad \alpha} + {c_{0}\sin \quad {\beta cos}\quad \alpha} + {a_{1i}\cos \quad \beta}}\quad} \\{{{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}}\quad} \\{{{- a_{0}}\sin \quad \beta} + {b_{0}\cos \quad {\beta sin}\quad \alpha} + {c_{0}\cos \quad {\beta cos}\quad \alpha} - {a_{1i}\sin \quad \beta}} \\{1\quad}\end{bmatrix}}\end{matrix} & (26) \\\begin{matrix}{P_{{susp}.n}^{r} = \quad {{\,_{4n}^{r}T}{{}_{7n}^{4n}{}_{{susp}.n}^{7n}}}} \\{= \quad {\begin{bmatrix}{\cos \quad \beta} & {\sin \quad {{\beta sin}\left( {\alpha + \gamma_{n}} \right)}} & {\sin \quad {{\beta cos}\left( {\alpha + \gamma_{n}} \right)}} & {{b_{2n}\sin \quad {\beta sin\alpha}} + {a_{1n}\cos \quad \beta}} \\0 & {\cos \left( {\alpha + \gamma_{n}} \right)} & {- {\sin \left( {\alpha + \gamma_{n}} \right)}} & {b_{2n}\cos \quad \alpha} \\{{- \sin}\quad \beta} & {\cos \quad {{\beta sin}\left( {\alpha + \gamma_{n}} \right)}} & {\cos \quad {{\beta cos}\left( {\alpha + \gamma_{n}} \right)}} & {z_{0} + {b_{2n}\cos \quad {\beta sin\alpha}} - {a_{1n}\sin \quad \beta}} \\0 & 0 & 0 & 1\end{bmatrix} \cdot}} \\{\quad {\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \eta_{n}} & {{- \sin}\quad \eta_{n}} & {{- z_{6n}}\sin \quad \eta_{n}} \\0 & {\sin \quad \eta_{n}} & {\cos \quad \eta_{n}} & {c_{1n} + {z_{6n}\cos \quad \eta_{n}}} \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}}} \\{= \quad \begin{bmatrix}{{\left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} \sin \quad \beta} + {a_{1n}\cos \quad \beta}} \\{{{{- z_{6n}}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {c_{1n}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\cos \quad \alpha}}\quad} \\{{\left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} \cos \quad \beta} - {a_{1n}\sin \quad \beta}} \\{1\quad}\end{bmatrix}}\end{matrix} & (27) \\\begin{matrix}{P_{{arm}.n}^{r} = \quad {{\,_{4n}^{r}T}{{}_{10n}^{4n}{}_{{arm}.n}^{10n}}}} \\{= \quad {\begin{bmatrix}{\cos \quad \beta} & {\sin \quad {{\beta sin}\left( {\alpha + \gamma_{n}} \right)}} & {\sin \quad {{\beta cos}\left( {\alpha + \gamma_{n}} \right)}} & {{b_{2n}\sin \quad {\beta sin}\quad \alpha} + {a_{1n}\cos \quad \beta}} \\0 & {\cos \left( {\alpha + \gamma_{n}} \right)} & {- {\sin \left( {\alpha + \gamma_{n}} \right)}} & {b_{2n}\cos \quad \alpha} \\{{- \sin}\quad \beta} & {\cos \quad {\beta sin}\quad \left( {\alpha + \gamma_{n}} \right)} & {\cos \quad \beta \quad {\cos \left( {\alpha + \gamma_{n}} \right)}} & {z_{0} + {b_{2n}\cos \quad {\beta sin}\quad \alpha} - {a_{1n}\sin \quad \beta}} \\0 & 0 & 0 & 1\end{bmatrix} \cdot}} \\{\quad {\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \theta_{n}} & {{- \sin}\quad \theta_{n}} & {e_{3n}\cos \quad \theta_{n}} \\0 & {\sin \quad \theta_{n}} & {\cos \quad \theta_{n}} & {c_{2n} + {e_{1n}\sin \quad \theta_{n}}} \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}}} \\{= \quad \begin{bmatrix}{{\left\{ {{e_{1}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} \sin \quad \beta} + {a_{1n}\cos \quad \beta}} \\{{{e_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\cos \quad \alpha}}\quad} \\{{\left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} \cos \quad \beta} - {a_{1n}\sin \quad \beta}} \\{1\quad}\end{bmatrix}}\end{matrix} & (28) \\\begin{matrix}{P_{{wheel}.n}^{r} = \quad {{\,_{4n}^{r}T}{{}_{12n}^{4n}{}_{{wheel}.n}^{12n}}}} \\{= \quad \begin{bmatrix}{\cos \quad \beta} & {\sin \quad {{\beta sin}\left( {\alpha + \gamma_{n}} \right)}} & {\sin \quad \beta \quad {\cos \left( {\alpha + \gamma_{n}} \right)}} & {{b_{2n}\sin \quad {\beta sin\alpha}} + {a_{1n}\cos \quad \beta}} \\0 & {\cos \left( {\alpha + \gamma_{n}} \right)} & {- {\sin \left( {\alpha + \gamma_{n}} \right)}} & {b_{2n}\cos \quad \alpha} \\{{- \sin}\quad \beta} & {\cos \quad {{\beta sin}\left( {\alpha + \gamma_{n}} \right)}} & {\cos \quad {{\beta cos}\left( {\alpha + \gamma_{n}} \right)}} & {{b_{2n}\cos \quad {\beta sin\alpha}} - {a_{1n}\sin \quad \beta}} \\0 & 0 & 0 & 1\end{bmatrix}} \\{\quad {\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \left( {\theta_{n} + \zeta_{n}} \right)} & {- {\sin \left( {\theta_{n} + \zeta_{n}} \right)}} & {e_{3n}\cos \quad \theta_{n}} \\0 & {\sin \left( {\theta_{n} + \zeta_{n}} \right)} & {\cos \left( {\theta_{n} + \zeta_{n}} \right)} & {c_{2n} + {e_{3n}\sin \quad \theta_{n}}} \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}}} \\{= \quad \begin{bmatrix}{{{\left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} \sin \quad \beta} + {a_{1n}\cos \quad \beta}}\quad} \\{{{e_{3n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{2}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\cos \quad \alpha}}\quad} \\{z_{0} + {\left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} \cos \quad \beta} - {a_{1n}\sin \quad \beta}} \\{1\quad}\end{bmatrix}}\end{matrix} & (29) \\\begin{matrix}{P_{{touchpoint}.n}^{r} = \quad {{\,_{4n}^{r}T}{\,_{12n}^{4n}T}{{}_{13n}^{12n}{}_{{touchpoint}.n}^{13n}}}} \\{= \quad \begin{bmatrix}{\cos \quad \beta} & {\sin \quad \beta \quad {\sin \left( {\alpha + \gamma_{n}} \right)}} & {\sin \quad \beta \quad {\cos \left( {\alpha + \gamma_{n}} \right)}} & {{b_{2n}\sin \quad {\beta sin}\quad \alpha} + {a_{1n}\cos \quad \beta}} \\0 & {\cos \left( {\alpha + \gamma_{n}} \right)} & {- {\sin \left( {\alpha + \gamma_{n}} \right)}} & {b_{2n}\cos \quad \alpha} \\{{- \sin}\quad \beta} & {\cos \quad {{\beta sin}\left( {\alpha + \gamma_{n}} \right)}} & {\cos \quad {{\beta cos}\left( {\alpha + \gamma_{n}} \right)}} & {z_{0} + {b_{2n}\cos \quad {\beta sin}\quad \alpha} - {a_{1n}\sin \quad \beta}} \\0 & 0 & 0 & 1\end{bmatrix}} \\{\quad {{\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \left( {\theta_{n} + \zeta_{n}} \right)} & {- {\sin \left( {\theta_{n} + \zeta_{n}} \right)}} & {e_{3n}\cos \quad \theta_{n}} \\0 & {\sin \left( {\theta_{n} + \zeta_{n}} \right)} & {\cos \left( {\theta_{n} + \zeta_{n}} \right)} & {c_{2n} + {e_{3n}\sin \quad \theta_{n}}} \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & z_{12n} \\0 & 0 & 0 & 1\end{bmatrix}}\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}}} \\{= \quad \begin{bmatrix}{{{\left\{ {{z_{12n}\cos \quad \alpha} + {e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} \sin \quad \beta} + {a_{1n}\cos \quad \beta}}\quad} \\{{{{- z_{12n}}\sin \quad \alpha} + {e_{3n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\cos \quad \alpha}}\quad} \\{z_{0} + {\left\{ {{z_{12n}\cos \quad \alpha} + {e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} \cos \quad \beta} - {a_{1n}\sin \quad \beta}} \\{1\quad}\end{bmatrix}}\end{matrix} & (30)\end{matrix}$

where ζ_(n) is substituted by,

ζ_(n)=−γ_(n)−θ_(n)

because of the link mechanism to support a wheel at this geometricrelation.

2.3 Description of the stabilizer linkage point in the local coordinatesystem {1 n}.

The stabilizer works as a spring in which force is proportional to thedifference of displacement between both arms in a local coordinatesystem {1 n} fixed to the body 310. $\begin{matrix}\begin{matrix}{P_{{stab}.n}^{1n} = \quad {{\,_{3n}^{1n}T}{\,_{4n}^{3n}T}{\,_{8n}^{4n}T}{\,_{9n}^{8n}T}{{}_{14n}^{9n}{}_{{stab}.n}^{14n}}}} \\{= \quad {{\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & b_{2n} \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \gamma_{n}} & {{- \sin}\quad \gamma_{n}} & 0 \\0 & {\sin \quad \gamma_{n}} & {\cos \quad \gamma_{n}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & c_{2n} \\0 & 0 & 0 & 1\end{bmatrix}}} \\{\quad {{\begin{bmatrix}1 & 0 & 0 & 0 \\0 & {\cos \quad \theta_{n}} & {{- \sin}\quad \theta_{n}} & 0 \\0 & {\sin \quad \theta_{n}} & {\cos \quad \theta_{n}} & 0 \\0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & e_{0n} \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}}\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}}} \\{= \quad \begin{bmatrix}{0\quad} \\{{e_{0n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}} - {c_{2}\sin \quad \gamma_{n}} + b_{2n}} \\{{{e_{0n}{\sin \left( {\gamma_{n} + \theta_{n}} \right)}} + {c_{2n}\cos \quad \gamma_{n}}}\quad} \\{0\quad}\end{bmatrix}}\end{matrix} & (31)\end{matrix}$

3. Kinetic energy, potential energy and dissipative functions for the<Body>, <Suspension>, <Arm>, <Wheel>and <Stabilizer>.

Kinetic energy and potential energy except by springs are calculatedbased on the displacement referred to the inertial global coordinate{r}. Potential energy by springs and dissipative functions arecalculated based on the movement in each local coordinate.

<Body>

T_(b) ^(tr)=1/2m_(b)({dot over (x)}_(b) ²+{dot over (y)}_(b) ²+{dot over(z)}_(b) ²)  (32)

where

x_(b)=(a₀+a_(1n))cos β6+(b₀ sin α+c₀ cos α)sin β

y_(b)=b₀cos α−c₀ sin α

z_(b)=z₀−(a₀+a_(1n))sin β+(b₀ sin α+c₀ cos α)cos β  (33)

and $\begin{matrix}\begin{matrix}{{q_{j,k} = \beta},\alpha,z_{0}} \\{\frac{\partial x_{b}}{\partial\beta} = {{{- \left( {a_{0} + a_{1n}} \right)}\sin \quad \beta} + {\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)\cos \quad \beta}}} \\{\frac{\partial x_{b}}{\partial\alpha} = {\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)\sin \quad \beta}} \\{\frac{\partial y_{b}}{\partial\beta} = {\frac{\partial x_{b}}{\partial z_{0}} = {\frac{\partial y_{b}}{\partial z_{0}} = 0}}} \\{\frac{\partial y_{b}}{\partial\alpha} = {{{- b_{0}}\sin \quad \alpha} - {c_{0}\cos \quad \alpha}}} \\{\frac{\partial z_{b}}{\partial\beta} = {{{- \left( {a_{0} + a_{1n}} \right)}\cos \quad \beta} - {\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)\sin \quad \beta}}} \\{\frac{\partial z_{b}}{\partial\alpha} = {\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)\cos \quad \beta}} \\{\frac{\partial z_{b}}{\partial z_{0}} = 1}\end{matrix} & (34)\end{matrix}$

and thus $\begin{matrix}\begin{matrix}{T_{b}^{tr} = \quad {\frac{1}{2}{m_{b}\left( {{\overset{.}{x}}_{b}^{2} + {\overset{.}{y}}_{b}^{2} + {\overset{.}{z}}_{b}^{2}} \right)}}} \\{= \quad {\frac{1}{2}m_{b}{\sum\limits_{j,k}\left( {{\frac{\partial x_{b}}{\partial q_{j}}\frac{\partial x_{b}}{\partial q_{k}}{\overset{.}{q}}_{j}{\overset{.}{q}}_{k}} + {\frac{\partial y_{b}}{\partial q_{j}}\frac{\partial y_{b}}{\partial q_{k}}{\overset{.}{q}}_{j}{\overset{.}{q}}_{k}} + {\frac{\partial z_{b}}{\partial q_{j}}\frac{\partial z_{b}}{\partial q_{k}}{\overset{.}{q}}_{j}{\overset{.}{q}}_{k}}} \right)}}} \\{= \quad {\frac{1}{2}m_{b}{\langle{{{\overset{.}{\beta}}^{2}\left\{ {{{- \left( {a_{0} + a_{1}} \right)}\sin \quad \beta} + {\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)\cos \quad \beta}} \right\}^{2}} +}}}} \\{\quad {{{\overset{.}{\alpha}}^{2}\left\{ {\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)\sin \quad \beta} \right\}^{2}} + {{\overset{.}{\alpha}}^{2}\left( {{{- b_{0}}\sin \quad \alpha} - {c_{0}\cos \quad \alpha}} \right)}^{2} +}} \\{\quad {{{\overset{.}{\beta}}^{2}\left\{ {{{- \left( {a_{0} + a_{1}} \right)}\cos \quad \beta} - {\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)\sin \quad \beta}} \right\}^{2}} +}} \\{\quad {{{\overset{.}{\alpha}}^{2}\left\{ {\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)\cos \quad \beta} \right\}^{2}} + {\overset{.}{z}}_{0}^{2} + {2\overset{.}{\alpha}\overset{.}{\beta}\left\lbrack \left\{ - \right. \right.}}} \\{{\quad \left. {{\left( {a_{0} + a_{1}} \right)\sin \quad \beta} + {\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)\cos \quad \beta}} \right\}}\left( {{b_{0}\cos \quad \alpha} -} \right.} \\{{{\quad \left. {c_{0}\sin \quad \alpha} \right)}\sin \quad \beta} + \left\{ {{{- \left( {a_{0} + a_{1}} \right)}\cos \quad \beta} - \left( {{b_{0}\sin \quad \alpha} +} \right.} \right.} \\{\left. {\left. {{\quad \left. {c_{0}\cos \quad \alpha} \right)}\sin \quad \beta} \right\} \left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)\cos \quad \beta} \right\rbrack -} \\{\quad {{2\overset{.}{\beta}{\overset{.}{z}}_{0}\left\{ {{\left( {a_{0} + a_{1n}} \right)\quad \cos \quad \beta} + {\left( {{b_{0}\sin \quad \alpha} - {c_{0}\cos \quad \alpha}} \right)\sin \quad \beta}} \right\}} +}} \\{\quad {{2\overset{.}{\alpha}{\overset{.}{z}}_{0}\left( {b_{0}\cos \quad \alpha} \right.} - {\left. c_{0}\sin \quad \alpha \right)\cos \quad \beta\rangle}}} \\{= \quad {\frac{1}{2}m_{b}{\langle{{{\overset{.}{\alpha}}^{2}\left( {b_{0}^{2} + c_{0}^{2}} \right)} + {{\overset{.}{\beta}}^{2}\left\{ {\left( {a_{0} + a_{1i}} \right)^{2} +} \right.}}}}} \\{{\quad \left. \left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)^{2} \right\}} + {\overset{.}{z}}_{0}^{2} - {2\quad \overset{¨}{\alpha}{\overset{.}{\beta}\left( {a_{0} + a_{1i}} \right)}\left( {{b_{0}\cos \quad \alpha} -} \right.}} \\{{\quad \left. {c_{0}\sin \quad \alpha} \right)} - {2\overset{.}{\beta}{\overset{.}{z}}_{0}\left\{ {{\left( {a_{0} + a_{1i}} \right)\cos \quad \beta} + \left( {{b_{0}\sin \quad \alpha} -} \right.} \right.}} \\{{{{\quad \left. {c_{0}\cos \quad \alpha} \right)}\sin \quad \beta} + {2\overset{.}{\alpha}{{\overset{.}{z}}_{0}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}\cos \quad \beta}}\rangle}\end{matrix} & (35)\end{matrix}$

T_(b) ^(ro)=1/2(I_(bx)ω_(bx) ²+I_(by)ω_(by) ²+I_(bz)ω_(bz) ²)

where

ω_(bx)={dot over (α)}

ω_(by)={dot over (β)}

ω_(bz)=0

$\begin{matrix}\begin{matrix}{T_{b}^{ro} = {\frac{1}{2}\left( {{I_{bx}{\overset{.}{\alpha}}^{2}} + {I_{by}{\overset{.}{\beta}}^{2}}} \right)}} \\{U_{b} = {m_{b}{gz}_{b}}} \\{= {m_{b}g\left\{ {{{- \left( {a_{0} + a_{1n}} \right)}\sin \quad \beta} + {\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)\cos \quad \beta}} \right\}}}\end{matrix} & (36)\end{matrix}$

<Suspension>

T_(sn) ^(tr)=1/2m_(sn)({dot over (x)}_(sn) ²+{dot over (y)}_(sn) ²+{dotover (z)}_(sn) ²)

where

x_(sn)={z_(6n) cos(α+γ_(n)+η_(n))+c_(1n) cos(α+γ_(n))+b_(2n) sin α}sinβ+a_(1n) cos β

y_(sn)=−z_(6f) sin(α+γ_(n)+η_(n))−c_(1n) sin(α+γ_(n))+b_(2n) cos α

z_(sn)=z₀+{z_(6n) cos(α+γ_(n)+η_(n))+c_(1n) cos(α+γ_(n))+b_(2n) sinα}cos β−a_(1n) sin β  (37)

$\begin{matrix}\begin{matrix}{{q_{j,k} = \quad z_{6n}},\eta_{n},\alpha,\beta,z_{0}} \\{\frac{\partial x_{sn}}{\partial z_{6n}} = \quad {{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\sin \quad \beta}} \\{\frac{\partial x_{sn}}{\partial\eta_{n}} = \quad {{- z_{6n}}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\sin \quad \beta}} \\{\frac{\partial x_{sn}}{\partial\alpha} = \quad {\left\{ {{{- z_{6n}}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {c_{1n}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\cos \quad \alpha}} \right\} \sin \quad \beta}} \\{\frac{\partial x_{sn}}{\partial\beta} = \quad \left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.} \\{{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\cos \quad \beta} - {a_{1n}\sin \quad \beta}} \\{\frac{\partial y_{sn}}{\partial z_{6n}} = \quad {- {\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}}} \\{\frac{\partial y_{sn}}{\partial\eta_{n}} = \quad {{- z_{6n}}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}}} \\{\frac{\partial y_{sn}}{\partial\alpha} = \quad {{{- z_{6n}}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} - {b_{2n}\sin \quad \alpha}}} \\{\frac{\partial y_{sn}}{\partial\beta} = \quad {\frac{\partial x_{sn}}{\partial z_{0}} = {\frac{\partial y_{sn}}{\partial z_{0}} = 0}}} \\{\frac{\partial z_{sn}}{\partial z_{0}} = \quad 1}\end{matrix} & (38) \\\begin{matrix}{\frac{\partial z_{sn}}{\partial z_{6n}} = \quad {{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta}} \\{\frac{\partial z_{sn}}{\partial\eta_{n}} = \quad {{- z_{6n}}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta}} \\{\frac{\partial z_{sn}}{\partial\alpha} = \quad \left\{ {{{- z_{6n}}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {c_{1n}{\sin \left( {\alpha + \gamma_{n}} \right)}} +} \right.} \\{{\quad \left. {b_{2n}\cos \quad \alpha} \right\}}\cos \quad \beta} \\{\frac{\partial z_{sn}}{\partial\beta} = \quad {- \left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\sin \quad \beta} - {a_{1n}\cos \quad \beta}}\end{matrix} & (39) \\\begin{matrix}{{\therefore\quad T_{sn}^{tr}} = \quad {\frac{1}{2}{m_{sn}\left( {{\overset{.}{x}}_{sn}^{2} + {\overset{.}{y}}_{sn}^{2} + {\overset{.}{z}}_{sn}^{2}} \right)}}} \\{= \quad {\frac{1}{2}m_{sn}{\sum\limits_{j,k}\left( {{\frac{\partial x_{sn}}{\partial q_{j}}\frac{\partial x_{sn}}{\partial q_{k}}{\overset{.}{q}}_{j}{\overset{.}{q}}_{k}} + {\frac{\partial y_{sn}}{\partial q_{j}}\frac{\partial y_{sn}}{\partial q_{k}}{\overset{.}{q}}_{j}{\overset{.}{q}}_{k}} +} \right.}}} \\{\quad \left. {\frac{\partial z_{sn}}{\partial q_{j}}\frac{\partial z_{sn}}{\partial q_{k}}{\overset{.}{q}}_{j}{\overset{.}{q}}_{k}} \right)}\end{matrix} & (40) \\\begin{matrix}{\quad {= \quad {\frac{1}{2}m_{sn}{\langle{{\overset{.}{z}}_{6n}^{2} + {{\overset{.}{\eta}}_{n}^{2}z_{6n}^{2}} + {{\overset{.}{\alpha}}^{2}\left\lbrack {z_{6n}^{2} + c_{1n}^{2} + b_{2n}^{2} +} \right.}}}}}} \\{\quad {2\left\{ {{z_{6n}c_{1n}\cos \quad \eta_{n}} - {z_{6n}b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}} -} \right.}} \\{\left. {\quad \left. {c_{1n}b_{2n}\sin \quad \gamma_{n}} \right\}} \right\rbrack + {{\overset{.}{\beta}}^{2}\left\lbrack \left\{ \left( {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right. \right. \right.}} \\{\left. \left. \quad {{c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right) \right\}^{2} + \left. a_{1n}^{2} \right\rbrack + {\overset{.}{z}}_{0}^{2} +} \\{\quad {{2{\overset{.}{z}}_{6n}\overset{.}{\alpha}\left\{ {{c_{1n}\sin \quad \eta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} -}} \\{\quad {{2{\overset{.}{z}}_{6n}\overset{.}{\beta}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {2{\overset{.}{\eta}}_{n}\overset{.}{\alpha}z_{6n}\left\{ {z_{6n} + {c_{1n}\cos \quad \eta_{n}} -} \right.}}} \\{{\quad \left. {b_{2}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}} \right\}} + {2{\overset{.}{\eta}}_{n}\overset{.}{\beta}z_{6n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \\{\quad {{2\overset{.}{\alpha}\overset{.}{\beta}a_{1n}\left\{ {{z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {b_{2n}\cos \quad \alpha}} \right\}} +}} \\{\quad {{2{\overset{.}{z}}_{6n}{\overset{.}{z}}_{0}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} - {2{\overset{.}{\eta}}_{n}{\overset{.}{z}}_{0}z_{6n}\sin\left( {\alpha + \gamma_{n} +} \right.}}} \\{{{\quad \left. \eta_{n} \right)}\cos \quad \beta} + {2\overset{.}{\alpha}{\overset{.}{z}}_{0}\left\{ {{z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {c_{1n}{\sin \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{\quad \left. {b_{2n}\cos \quad \alpha} \right\}}\cos \quad \beta} + {2\overset{.}{\beta}{\overset{.}{z}}_{0}\left\lbrack \left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right. \right.}} \\{\left. {{{\quad \left. {{c_{1n}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\cos \quad \alpha}} \right\}}\sin \quad \beta} + {\alpha_{1n}\cos \quad \beta}} \right\rbrack\rangle}\end{matrix} & (41) \\\begin{matrix}{T_{sn}^{ro} \cong \quad 0} \\{U_{sn} = \quad {{m_{sn}{gz}_{sn}} + {\frac{1}{2}{k_{sn}\left( {z_{6n} - l_{sn}} \right)}^{2}}}} \\{= \quad {m_{sn}g\left\lbrack {z_{0} + \left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.} \right.}} \\{\left. {{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\cos \quad \beta} - {a_{1n}\sin \quad \beta}} \right\rbrack + {\frac{1}{2}{k_{sn}\left( {z_{6n} - l_{sn}} \right)}^{2}}} \\{F_{sn} = \quad {{- \frac{1}{2}}c_{sn}{\overset{.}{z}}_{6n}^{2}}}\end{matrix} & (42) \\{\langle{Arm}\rangle} & (43) \\{T_{an}^{tr} = {\frac{1}{2}{m_{an}\left( {{\overset{.}{x}}_{an}^{2} + {\overset{.}{y}}_{an}^{2} + {\overset{.}{z}}_{an}^{2}} \right)}}} & \quad\end{matrix}$

where

x_(an)={e_(1n) sin(α+γ_(n)+θ_(n))+c_(2n) cos(α+γ_(n))+b_(2n) sin α}sinβ+a_(1n) cos β

y_(an)=e_(1n) cos(α+γ_(n)+θ_(n))−c_(2n) sin(α+γ_(n))+b_(2n) cos α

z_(an)=z₀+{e_(1n) sin(α+γ_(n)+θ_(n))+c_(2n) cos(α+γ_(n))+b_(2n) sinα}cos β−a_(1n) sin β  (44)

and $\begin{matrix}\begin{matrix}{{q_{j,k} = \quad \theta_{n}},\alpha,\beta,z_{0}} \\{\frac{\partial x_{an}}{\partial\theta_{n}} = \quad {e_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\sin \quad \beta}} \\{\frac{\partial x_{an}}{\partial\alpha} = \quad \left\{ {{e_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} +} \right.} \\{{\quad \left. {b_{2n}\cos \quad \alpha} \right\}}\sin \quad \beta} \\{\frac{\partial x_{an}}{\partial\beta} = \quad \left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.} \\{{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\cos \quad \beta} - {a_{1n}\sin \quad \beta}} \\{\frac{\partial y_{an}}{\partial\theta_{n}} = \quad {{- e_{1n}}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}}} \\{\frac{\partial y_{an}}{\partial\alpha} = \quad {{{- e_{1n}}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} - {b_{2n}\sin \quad \alpha}}} \\{\frac{\partial y_{cn}}{\partial\beta} = \quad {\frac{\partial x_{an}}{\partial z_{0}} = {\frac{\partial y_{an}}{\partial z_{0}} = 0}}} \\{\frac{\partial z_{an}}{\partial\theta_{n}} = \quad {e_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\cos \quad \beta}} \\{\frac{\partial z_{an}}{\partial\alpha} = \quad \left\{ {{e_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} +} \right.} \\{{\quad \left. {b_{2n}\cos \quad \alpha} \right\}}\cos \quad \beta} \\{\frac{\partial z_{an}}{\partial\beta} = \quad {- \left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\sin \quad \beta} - {a_{1n}\cos \quad \beta}} \\{\frac{\partial z_{an}}{\partial z_{0}} = \quad 1}\end{matrix} & (45)\end{matrix}$

thus $\begin{matrix}\begin{matrix}{T_{an}^{tr} = \quad {\frac{1}{2}{m_{an}\left( {{\overset{.}{x}}_{an}^{2} + {\overset{.}{y}}_{an}^{2} + {\overset{.}{z}}_{an}^{2}} \right)}}} \\{= \quad {\frac{1}{2}m_{an}{\sum\limits_{j,k}\left( {{\frac{\partial x_{an}}{\partial q_{j}}\frac{\partial x_{an}}{\partial q_{k}}{\overset{.}{q}}_{j}{\overset{.}{q}}_{k}} + {\frac{\partial y_{an}}{\partial q_{j}}\frac{\partial y_{an}}{\partial q_{k}}{\overset{.}{q}}_{j}{\overset{.}{q}}_{k}} +} \right.}}} \\{\quad \left. {\frac{\partial z_{an}}{\partial q_{j}}\frac{\partial z_{an}}{\partial q_{k}}{\overset{.}{q}}_{j}{\overset{.}{q}}_{k}} \right)}\end{matrix} & (46) \\\begin{matrix}{= \quad {\frac{1}{2}m_{an}{\langle{{{\overset{.}{\theta}}_{n}^{2}e_{1n}^{2}} + {{\overset{.}{\alpha}}^{2}\left\lbrack {e_{1n}^{2} + c_{2n}^{2} + b_{2n}^{2} - {2\left\{ {{e_{1n}c_{2n}\sin \quad \theta_{n}} +} \right.}} \right.}}}}} \\{\left. {\quad \left. {{e_{1n}b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}} + {c_{2n}b_{2n}\sin \quad \gamma_{n}}} \right\}} \right\rbrack +} \\{\quad {{\overset{.}{\beta}}^{2}\left\lbrack {\left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}^{2} +} \right.}} \\{{\quad \left. a_{1n}^{2} \right\rbrack} + {\overset{.}{z}}_{0}^{2} + {2\overset{.}{\theta}\overset{.}{\alpha}e_{1n}\left\{ {e_{1n} - {c_{2n}\sin \quad \theta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}} -} \\{\quad {{2{\overset{.}{\theta}}_{n}\overset{.}{\beta}e_{1n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {2\overset{.}{\alpha}\overset{.}{\beta}a_{1n}\left\{ {{e_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -} \right.}}} \\{{\quad \left. {{c_{1n}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\cos \quad \alpha}} \right\}} - {2{\overset{.}{\theta}}_{n}{\overset{.}{z}}_{0}e_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\cos \quad \beta} +} \\{\quad {2\overset{.}{\alpha}{\overset{.}{z}}_{0}\left\{ {{e_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{\quad \left. {b_{2n}\cos \quad \alpha} \right\}}\cos \quad \beta} + {2\overset{.}{\beta}{\overset{.}{z}}_{0}\left\lbrack \left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +}\quad \right. \right.}} \\{\left. {{{\quad \left. {{c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}}\sin \quad \beta} + {\alpha_{1n}\cos \quad \beta}} \right\rbrack\rangle}\end{matrix} & (47) \\\begin{matrix}{T_{an}^{ro} = \quad {\frac{1}{2}I_{ax}\omega_{ax}^{2}}} \\{= \quad {\frac{1}{2}{I_{ax}\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)}^{2}}} \\{U_{an} = \quad {m_{an}{gz}_{an}}} \\{= \quad {m_{an}g\left\lbrack {z_{0} + \left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.} \right.}} \\\left. {{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\cos \quad \beta} - {a_{1n}\sin \quad \beta}} \right\rbrack\end{matrix} & (48)\end{matrix}$

<Wheel>

T_(wn) ^(tr)=1/2m_(wn)({dot over (x)}_(wn) ²+{dot over (y)}_(wn) ²+{dotover (z)}_(wn) ²)  (49)

where

x_(wn)={e_(3n) sin(α+γ_(n)+θ_(n))+c_(2n) cos(α+γ_(n))+b_(2n) sin α}sinβ+a_(1n) cos β

y_(wn)=e_(3n) cos(α+γ_(n)+θ_(n))−c_(2n) sin(α+γ_(n))+b_(2n) cos α

z_(wn)=z₀+{e_(3n) sin(α+γ_(n)+θ_(n))+c_(2n) cos(α+γ_(n))+b_(2n) sinα}cos β−a_(1n) sin β  (50)

Substituting m_(an) with m_(wn) and e_(1n) with e_(3n) in the equationfor the arm, yields an equation for the wheel as: $\begin{matrix}\begin{matrix}{T_{wn}^{tr} = \quad {\frac{1}{2}m_{wn}{\langle{{{\overset{.}{\theta}}_{n}^{2}e_{3n}^{2}} + {{\overset{.}{\alpha}}^{2}\left\lbrack {e_{3n}^{2} + c_{2n}^{2} + b_{2n}^{2} - {2\left\{ {{e_{3n}c_{2n}\sin \quad \theta_{n}} +} \right.}} \right.}}}}} \\{\left. {\quad \left. {{e_{3n}b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}} + {c_{2n}b_{2n}\sin \quad \gamma_{n}}} \right\}} \right\rbrack +} \\{\quad {{\overset{.}{\beta}}^{2}\left\lbrack \left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right. \right.}} \\{\left. {{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}^{2} + a_{1n}^{2}} \right\rbrack + {\overset{.}{z}}_{0}^{2} + {2\overset{.}{\theta}\overset{.}{\alpha}e_{3n}\left\{ {e_{3n} - {c_{2n}\sin \quad \theta_{n}} +} \right.}} \\{{\quad \left. {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}} \right\}} - {2{\overset{.}{\theta}}_{n}\overset{.}{\beta}e_{3n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -} \\{\quad {2\overset{.}{\alpha}\overset{.}{\beta}a_{1n}\left\{ {{e_{3n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{1n}{\sin \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{\quad \left. {b_{2n}\cos \quad \alpha} \right\}} + {2{\overset{.}{\theta}}_{n}{\overset{.}{z}}_{0}e_{3n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\cos \quad \beta} +} \\{\quad {2\overset{.}{\alpha}{\overset{.}{z}}_{0}\left\{ {{e_{3n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{\quad \left. {b_{2n}\cos \quad \alpha} \right\}}\cos \quad \beta} - {2\overset{.}{\beta}{\overset{.}{z}}_{0}\left\lbrack \left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right. \right.}} \\{\left. {{{\quad \left. {{c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}}\sin \quad \beta} + {\alpha_{1n}\cos \quad \beta}} \right\rbrack\rangle}\end{matrix} & (51) \\\begin{matrix}{T_{wn}^{ro} = \quad 0} \\{U_{wn} = \quad {{m_{wn}{gz}_{wn}} + {\frac{1}{2}{k_{wn}\left( {z_{12n} - l_{wn}} \right)}^{2}}}} \\{= \quad {m_{wn}g\left\lbrack {z_{0} + \left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.} \right.}} \\{\left. {{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\cos \quad \beta} - {a_{1n}\sin \quad \beta}} \right\rbrack + {\frac{1}{2}{k_{wn}\left( {z_{12n} - l_{wn}} \right)}^{2}}} \\{F_{wn} = \quad {{- \frac{1}{2}}c_{wn}{\overset{.}{z}}_{12n}^{2}}}\end{matrix} & (52)\end{matrix}$

<Stabilizer>

T_(zn) ^(tr)≅0  (53)

T_(zn) ^(ro)≅0  (54)

U_(zi,ii)=1/2k_(zi)(z_(zi)−z_(zii))²=1/2k_(zi)[{e_(0i)sin(γ_(i)+θ_(i))+c_(2i) cos γ_(i)}−{e_(0ii) sin(γ_(ii)+θ_(ii))+c_(2ii)cos γ_(ii)}]²=1/2k_(zi)e_(0i) ²{sin(γ_(i)+θ_(i))+sin(γ_(ii)+θ_(ii))}²

where

e_(0ii)=−e_(0i), c_(2ii)=c_(2ii), γ_(ii)=−γ_(i)=

U_(ziii,iv)≅1/2k_(ziii) (z_(ziii)−z_(ziv))²=1/2k_(ziii)[{e_(0iii)sin(γ_(iii)+θ_(iii))+c_(2iii) cos γ_(iii)}−{e_(0iv)sin(γ_(iv)+θ_(iv))+c_(2iv) cos γ_(iv)}]²=1/2k_(ziii)e_(0iii)²{sin(γ_(iii)+θ_(iii))+sin(γ_(iv)+θ_(iiv))}²

where

e_(0iii)=−e_(0iii c) _(2iv)=c_(2iii), γ_(iv)=−γiii  (55)

F_(zn)≅0  (56)

Therefore the total kinetic energy is: $\begin{matrix}{T_{tot} = {T_{b}^{tr} + {\sum\limits_{n = i}^{iv}\quad {{T_{sn}^{tr} + T_{an}^{tr} + T_{wn}^{tr} + T_{b}^{ro} + T_{an}^{ro}}}}}} & (57) \\\begin{matrix}{T_{tot} = \quad {T_{b}^{tr} + {\sum\limits_{n = i}^{iv}\quad {{T_{sn}^{tr} + T_{an}^{tr} + T_{wn}^{tr} + T_{b}^{ro} + T_{an}^{ro}}}}}} \\{= \quad {\frac{1}{2}m_{b}{\langle{{{\overset{.}{\alpha}}^{2}\left( {b_{0}^{2} + c_{0}^{2}} \right)} + {{\overset{.}{\beta}}^{2}\left\{ {\left( {a_{0} + a_{1i}} \right)^{2} +} \right.}}}}} \\{{\quad \left. \left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)^{2} \right\}} + {\overset{.}{z}}_{0}^{2} - {2\overset{.}{\alpha}{\overset{.}{\beta}\left( {a_{0} + a_{1i}} \right)}b_{0}\cos \quad \alpha} -} \\{{\quad \left. {c_{0}\sin \quad \alpha} \right)} - {2\overset{.}{\beta}{\overset{.}{z}}_{0}\left\{ {{\left( {a_{0} + a_{1i}} \right)\cos \quad \beta} + \left( {{b_{0}\sin \quad \alpha} +} \right.} \right.}} \\{{{\left. {{\quad \left. {c_{0}\cos \quad \alpha} \right)}\sin \quad \beta} \right\} + {2\overset{.}{\alpha}{{\overset{.}{z}}_{0}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}\cos \quad \beta}}\rangle} +} \\{\quad {\sum\limits_{n = i}^{iv}\quad {{\frac{1}{2}m_{sn}{\langle{{\overset{.}{z}}_{6n}^{2} + {{\overset{.}{\eta}}_{n}^{2}z_{6n}^{2}} + {{\overset{.}{\alpha}}^{2}\left\lbrack {z_{6n}^{2} + c_{1n}^{2} + b_{2n}^{2} +} \right.}}}}}}} \\{{\quad \left. {2\left\{ {{z_{6n}c_{1n}\cos \quad \eta_{n}} - {z_{6n}b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}} - {c_{1n}b_{2n}\sin \quad \gamma_{n}}} \right\}} \right\rbrack} +} \\{\quad {{\overset{.}{\beta}}^{2}\left\lbrack \left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right. \right.}} \\{\left. {{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}^{2} + a_{1n}^{2}} \right\rbrack + {\overset{.}{z}}_{0}^{2} + {2{\overset{.}{z}}_{6n}\overset{.}{\alpha}\left\{ {{c_{1n}\sin \quad \eta_{n}} +} \right.}} \\{{\quad \left. {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}} \right\}} - {2{\overset{.}{z}}_{6n}\overset{.}{\beta}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \\{\quad {{2{\overset{.}{\eta}}_{n}\overset{.}{\alpha}z_{6n}\left\{ {z_{6n} + {c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{2{\overset{.}{\eta}}_{n}\overset{.}{\beta}z_{6n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {2\overset{.}{\alpha}\overset{.}{\beta}a_{1n}\left\{ {z_{6n}{\sin\left( {\alpha +} \right.}} \right.}}} \\{{\quad \left. {\gamma_{n} + \eta_{n}} \right)} + {c_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\sin \left( {\alpha + \gamma_{n}} \right)}} -} \\{{\quad \left. {b_{2n}\cos \quad \alpha} \right\}} + {2{\overset{.}{z}}_{6n}{\overset{.}{z}}_{0}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} -} \\{\quad {{2{\overset{.}{\eta}}_{n}{\overset{.}{z}}_{0}z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} + {2\overset{.}{\alpha}{\overset{.}{z}}_{0}\left\{ {{- z_{6n}}{\sin\left( {\alpha +} \right.}} \right.}}} \\{{\left. {{\quad \left. {\gamma_{n} + \eta_{n}} \right)} - {c_{1n}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\cos \quad \alpha}} \right\} \cos \quad \beta} -} \\{\quad {2\overset{.}{\beta}{{\overset{.}{z}}_{0}\left\lbrack \left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right. \right.}}} \\{{\left. {{\quad \left. {b_{2n}\sin \quad \alpha} \right\}} + {a_{1n}\cos \quad \beta}} \right\rbrack\rangle} + {\frac{1}{2}m_{an}{\langle{{{\overset{.}{\theta}}_{n}^{2}e_{1n}^{2}} + {{\overset{.}{\alpha}}^{2}\left\lbrack {e_{1n}^{2} +} \right.}}}}} \\{\quad {c_{2n}^{2} + b_{2n}^{2} - {2\left\{ {{e_{1n}c_{2n}\sin \quad \theta_{n}} + {e_{1n}b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}} +} \right.}}} \\{\left. {\quad \left. {c_{2n}b_{2n}\sin \quad \gamma_{n}} \right\}} \right\rbrack + {\overset{.}{\beta^{2}}\left\lbrack \left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right. \right.}} \\{\left. {{\quad \left. {{c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}}^{2} + a_{1n}^{2}} \right\rbrack + {\overset{.}{z}}_{0}^{2} + {2\overset{.}{\theta}\overset{.}{\alpha}e_{1n}\left\{ {e_{1n} -} \right.}} \\{{\quad \left. {{c_{2n}\sin \quad \theta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}} - {2{\overset{.}{\theta}}_{n}\overset{.}{\beta}e_{1n}a_{1n}{\cos\left( {\alpha + \gamma_{n} +} \right.}}} \\{{\quad \left. \theta_{n} \right)} - {2\overset{.}{\alpha}\overset{.}{\beta}a_{1n}\left\{ {{e_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{1n}{\sin \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{\quad \left. {b_{2n}\cos \quad \alpha} \right\}} + {2{\overset{.}{\theta}}_{n}{\overset{.}{z}}_{0}e_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\cos \quad \beta} +} \\{\quad {2\overset{.}{\alpha}{\overset{.}{z}}_{0}\left\{ {{e_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{\quad \left. {b_{2n}\cos \quad \alpha} \right\}}\cos \quad \beta} - {2\overset{.}{\beta}{{\overset{.}{z}}_{0}\left\lbrack \left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right. \right.}}} \\{{{{\quad \left. {{c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}} + {a_{1n}\cos \quad \beta}}\rangle} +} \\{\quad {\frac{1}{2}m_{wn}{\langle{{{\overset{.}{\theta}}_{n}^{2}e_{3n}^{2}} + {{\overset{.}{\alpha}}^{2}\left\lbrack {e_{3n}^{2} + c_{2n}^{2} + b_{2n}^{2} - {2\left\{ {{e_{3n}c_{2n}\sin \quad \theta_{n}} -} \right.}} \right.}}}}} \\{\left. {\quad \left. {{e_{3n}b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}} + {c_{2n}b_{2n}\sin \quad \gamma_{n}}} \right\}} \right\rbrack +} \\{\quad {{\overset{.}{\beta}}^{2}\left\lbrack {\left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}^{2} +} \right.}} \\{{\quad \left. a_{1n}^{2} \right\rbrack} + {\overset{.}{z}}_{0}^{2} + {2\overset{.}{\theta}\overset{.}{\alpha}e_{3n}\left\{ {e_{3n} - {c_{2n}\sin \quad \theta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}} -} \\{\quad {{2{\overset{.}{\theta}}_{n}\overset{.}{\beta}e_{3n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {2\overset{.}{\alpha}\overset{.}{\beta}a_{1n}\left\{ {e_{3n}{\cos\left( {\alpha + \gamma_{n} +} \right.}} \right.}}} \\{\left. {{\quad \left. \theta_{n} \right)} - {c_{1n}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\cos \quad \alpha}} \right\} + {2{\overset{.}{\theta}}_{n}{\overset{.}{z}}_{0}e_{3n}{\cos\left( {\alpha + \gamma_{n} +} \right.}}} \\{{{\quad \left. \eta_{n} \right)}\cos \quad \beta} + {2\overset{.}{\alpha}{\overset{.}{z}}_{0}\left\{ {{e_{3n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{\quad \left. {b_{2n}\cos \quad \alpha} \right\}}\cos \quad \beta} - {2\overset{.}{\beta}{{\overset{.}{z}}_{0}\left\lbrack \left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -} \right. \right.}}} \\{{{{\quad \left. {{c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}} + {a_{1n}\cos \quad \beta}}\rangle} +} \\{\quad {{\frac{1}{2}\left( {{I_{bx}{\overset{.}{\alpha}}^{2}} + {I_{by}{\overset{.}{\beta}}^{2}}} \right)} + {\frac{1}{2}{I_{anx}\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)}^{2}}}}\end{matrix} & (58) \\\begin{matrix}{= \quad {\frac{1}{2}\left\lbrack {{{\overset{.}{\alpha}}^{2}m_{b\quad b\quad l}} + {{\overset{.}{\beta}}^{2}\left\{ {m_{bal} + {m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}^{2}} \right\}} +} \right.}} \\{\quad {{{\overset{.}{z}}_{0}^{2}m_{b}} - {2{\overset{.}{\alpha}\left( {{\overset{.}{\beta}m_{ba}} - {{\overset{.}{z}}_{0}m_{b}\cos \quad \beta}} \right)}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)} -}} \\{{\quad \left. {2\overset{.}{\beta}{\overset{.}{z}}_{0}\left\{ {{m_{ba}\cos \quad \beta} + {{m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}\sin \quad \beta}} \right\}} \right\rbrack} +} \\{\quad {\frac{1}{2}{\sum\limits_{n = i}^{iv}\quad {{{m_{sn}\left( {{\overset{.}{z}}_{6n}^{2} + {{\overset{.}{\eta}}_{n}^{2}z_{6n}^{2}}} \right)} + {{\overset{.}{\theta}}_{n}^{2}m_{aw21n}} + {{\overset{.}{z}}_{0}^{2}m_{sawn}} +}}}}} \\{\quad {{\overset{.}{\alpha}}^{2}{\langle{m_{saw1n} + {m_{sn}z_{6n}\left\lbrack {z_{6n} + {2m_{sn}\left\{ {{c_{1n}\cos \quad \eta_{n}} - {b_{2n}\sin\left( {\gamma_{n} +} \right.}} \right.}} \right.}}}}} \\{{{\left. \left. {\quad \left. \eta_{n} \right)} \right\} \right\rbrack - {2m_{aw1n}\left\{ {{c_{2n}\sin \quad \theta_{n}} - {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}}}\rangle} +} \\{\quad {{\overset{.}{\beta}}^{2}{\langle{m_{saw2n} + {m_{sn}\left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}}}}} \\{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}^{2} + {m_{an}\left\{ {{e_{1}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}^{2} + {m_{wn}\left\{ {{e_{3}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}^{2}\rangle} + {2{\overset{.}{z}}_{6n}\overset{.}{\alpha}m_{sn}\left\{ {{\sin \quad \eta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} -} \\{\quad {{2{\overset{.}{z}}_{6n}\overset{.}{\beta}{ma}_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {2{\overset{.}{\eta}}_{n}\overset{.}{\alpha}m_{sn}z_{6n}\left\{ {z_{6n} + {c_{1n}\cos \quad \eta_{n}} -} \right.}}} \\{{\quad \left. {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}} \right\}} + {2{\overset{.}{\eta}}_{n}\overset{.}{\beta}m_{sn}z_{6n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \\{\quad {{2\overset{.}{\theta}{\overset{.}{\alpha}\left\lbrack {m_{aw21n} - {m_{aw1n}\left\{ {{c_{2}\sin \quad \theta_{n}} - {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}}} \right\rbrack}} -}} \\{\quad {{2\overset{.}{\theta}\overset{.}{\beta}m_{aw1n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {2\overset{.}{\alpha}\overset{.}{\beta}a_{1n}\left\{ {{m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} -} \right.}}} \\{\quad {{m_{sawbn}\cos \quad \alpha} + {m_{sn}z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {m_{aw1n}\cos\left( {\alpha + \gamma_{n} +} \right.}}} \\{\left. {\quad \left. \theta_{n} \right)} \right\} + {2{\overset{.}{z}}_{6n}{\overset{.}{z}}_{0}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} -} \\{\quad {{2\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right){\overset{.}{z}}_{0}z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} + {2{\overset{.}{\theta}}_{n}{\overset{.}{z}}_{0}m_{awln}\cos\left( {\alpha +} \right.}}} \\{{{\quad \left. {\gamma_{n} + \theta_{n}} \right)}\cos \quad \beta} + {2\overset{.}{\alpha}{\overset{.}{z}}_{0}\left\{ {{m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {m_{sawcn}\sin\left( {\alpha +} \right.}} \right.}} \\{{\left. {{\quad \left. \gamma_{n} \right)} + {m_{sawbn}\cos \quad \alpha}} \right\} \cos \quad \beta} - {2\overset{.}{\beta}{\overset{.}{z}}_{0}\left\lbrack \left\{ {{z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -} \right. \right.}} \\{\quad {{m_{awln}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} +}} \\{{\left. {{{\quad \left. {m_{sawbn}\sin \quad \alpha} \right\}}\sin \quad \beta} + {m_{sawan}\cos \quad \beta}} \right\rbrack\rangle}}\end{matrix} & (59)\end{matrix}$

where

m_(ba)=m_(b)(a₀+a_(1i))

m_(bbI)=m_(b)(b₀ ²+c₀ ²)+I_(bx)

m_(baI)=m_(b)(a₀+a_(1i))²+I_(by)

m_(sawn)=m_(sn)+m_(an)+m_(wn)

m_(sawan)=(m_(sn)+m_(an)+m_(wn))a_(1n)

m_(sawbn)=(m_(sn)+m_(an)+m_(wn))b_(2n)  (60)

m_(sawcn)=m_(5n)c_(1n)+(m_(an)+m_(wn))c_(2n)

m_(saw2n)=(m_(sn)+m_(an)+m_(wn))a_(1n) ²

m_(sawIn)m_(an)e_(1n) ²m_(wn)e_(3n) ²+m_(sn)(c_(1n) ²+b_(2n)²−2c_(1n)b_(2n) sin γ_(n))+(m_(an)+m_(wn))(c_(2n) ²+b_(2n)²−2c_(2n)b_(2n) sin γ_(n))+I_(axn)

m_(aw2In)=m_(an)e_(1n) ²+m_(wn)e_(3n) ²+I_(axn)

m_(aw1n)=m_(an)e_(1n)+m_(wn)e_(3n)

m_(aw2n)=m_(an)e_(1n) ²+m_(wn)e_(3n) ²

Hereafter variables and coefficients which have index “n” impliesimplicit or explicit that they require summation with n=i, ii, iii, andiv.

Total potential energy is: $\begin{matrix}{U_{tot} = \quad {U_{b} + {\sum\limits_{n = i}^{iv}\quad {{U_{sn} + U_{an} + U_{wn} + U_{zn}}}}}} & (61) \\\begin{matrix}{= \quad {{m_{b}g\left\{ {z_{0} - {\left( {a_{0} - a_{1n}} \right)\sin \quad \beta} + {\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)\cos \quad \beta}} \right\}} +}} \\{\quad {\sum\limits_{n = i}^{iv}\quad {{m_{sn}g\left\lbrack {z_{0} + \left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.} \right.}}}} \\{\left. {{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\cos \quad \beta} - {a_{1n}\sin \quad \beta}} \right\rbrack + {\frac{1}{2}{k_{sn}\left( {z_{6n} - l_{sn}} \right)}^{2}} + {m_{an}g\left\lbrack {z_{0} +} \right.}} \\{\quad {{\left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} \cos \quad \beta} -}} \\{{\quad \left. {a_{1n}\sin \quad \beta} \right\rbrack} + {m_{wn}g\left\lbrack {z_{0} + \left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right.} \right.}} \\{\left. {{{\quad \left. {{c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}}\cos \quad \beta} - {a_{1n}\sin \quad \beta}} \right\rbrack +} \\{{\quad {\frac{1}{2}{k_{wn}\left( {z_{12n} - l_{wn}} \right)}^{2}}} + {\frac{1}{2}k_{zi}e_{ei}^{2}\left\{ {{\sin \left( {\gamma_{i} + \theta_{i}} \right)} + {\sin \left( {\gamma_{ii} + \theta_{ii}} \right)}} \right\}^{2}} +} \\{\quad {\frac{1}{2}k_{ziii}e_{oiii}^{2}\left\{ {{\sin \left( {\gamma_{iii} + \theta_{iii}} \right)} + {\sin \left( {\gamma_{iv} + \theta_{iv}} \right)}} \right\}^{2}}}\end{matrix} & (62) \\\begin{matrix}{= \quad {{g\left\{ {{z_{0}m_{b}} - {m_{ba}\sin \quad \beta} + {{m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}\cos \quad \beta}} \right\}} +}} \\{\quad {\sum\limits_{n = i}^{iv}\quad {\langle{g\left\lbrack \left\{ {{z_{0}m_{sawn}} + {m_{sn}z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right. \right.}}}} \\{\quad {{m_{aw1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} +}} \\{\left. {{{\quad \left. {m_{sawbn}\sin \quad \alpha} \right\}}\cos \quad \beta} - {m_{sawan}\sin \quad \beta}} \right\rbrack + {\frac{1}{2}{k_{sn}\left( {z_{6n} - l_{sn}} \right)}^{2}} +} \\{{\quad {\frac{1}{2}{k_{wn}\left( {z_{12n} - l_{wn}} \right)}^{2}}\rangle} + {\frac{1}{2}k_{zi}e_{0i}^{2}\left\{ {{\sin \left( {\gamma_{i} + \theta_{i}} \right)} + {\sin \left( {\gamma_{ii} + \theta_{ii}} \right)}} \right\}^{2}} +} \\{\quad {\frac{1}{2}k_{ziii}e_{oiii}^{2}\left\{ {{\sin \left( {\gamma_{iii} + \theta_{iii}} \right)} + {\sin \left( {\gamma_{iv} + \theta_{iv}} \right)}} \right\}^{2}}}\end{matrix} & (63)\end{matrix}$

where

m_(ba)=m_(b)(a₀+a_(1i))

m_(sawan)=(m_(sn)+m_(an)+m_(wn))a_(1n)

m_(sawbn)=(m_(sn)+m_(an)+m_(wn))b_(2n)

m_(sawcn)=m_(sn)c_(1n)+(m_(an)+m_(wn))c_(2n)

γ_(ii)=−γ_(i)  (64)

4. Lagrange's Equation

The Lagrangian is written as: $\begin{matrix}\begin{matrix}{L = \quad {T_{tot} - U_{tot}}} \\{= \quad {\frac{1}{2}\left\lbrack {{{\overset{.}{\alpha}}^{2}m_{b\quad b\quad l}} + {{\overset{.}{\beta}}^{2}\left\{ {m_{bal} + {m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}^{2}} \right\}} + {{\overset{.}{z}}_{0}^{2}m_{b}} -} \right.}} \\{{\quad \left. {\left( {{2\overset{.}{\alpha}\overset{.}{\beta}m_{ba}} - {{\overset{.}{z}}_{0}m_{b}\cos \quad \beta}} \right)\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)} \right\rbrack} -} \\{{\quad \left. {2\overset{.}{\beta}{\overset{.}{z}}_{0}\left\{ {{m_{ba}\cos \quad \beta} + {{m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}\sin \quad \beta}} \right\}} \right\rbrack} +} \\{\quad \left. {\frac{1}{2}\sum\limits_{n = i}^{iv}} \middle| {{m_{sn}\left( {{\overset{.}{z}}_{6n}^{2} + {{\overset{.}{\eta}}_{n}^{2}z_{6n}^{2}}} \right)} + {{\overset{.}{\theta}}_{n}^{2}m_{aw2ln}} + {{\overset{.}{z}}_{0}^{2}m_{sawn}} +} \right.} \\{\quad {{\overset{.}{\alpha}}^{2}{\langle{m_{saw1n} + {m_{sn}{z_{6n}\left\lbrack {z_{6n} + {2\left\{ {{c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}}} \right\rbrack}} -}}}} \\{{\quad {{2m_{awln}\left\{ {{c_{2n}\sin \quad \theta_{n}} - {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}}\rangle}} +} \\{\quad {{\overset{.}{\beta}}^{2}{\langle{m_{saw2n} + {m_{sn}\left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right.}}}}} \\{{\quad \left. {{c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}}^{2} + {m_{an}\left\{ {{e_{1}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right.}} \\{{\quad \left. {{c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}}^{2} + {m_{wn}\left\{ {{e_{3}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right.}} \\{{{\quad \left. {{c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}}^{2}\rangle} +} \\{\quad {{2{\overset{.}{z}}_{6n}\overset{.}{\alpha}m_{sn}\left\{ {{c_{1n}\sin \quad \eta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} -}} \\{\quad {{2{\overset{.}{z}}_{6n}\overset{.}{\beta}m_{sn}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{2{\overset{.}{\eta}}_{n}\overset{.}{\alpha}m_{sn}z_{6n}\left\{ {z_{6n} + {c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{2{\overset{.}{\eta}}_{n}\overset{.}{\beta}m_{sn}z_{6n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{2\overset{.}{\theta}{\overset{.}{\alpha}\left\lbrack {m_{aw2ln} - {m_{aw1n}\left\{ {{c_{2n}\sin \quad \theta_{n}} - {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}}} \right\rbrack}} -}} \\{\quad {{2\overset{.}{\theta}\overset{.}{\beta}m_{aw1n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +}} \\{\quad {2\overset{.}{\alpha}\overset{.}{\beta}a_{1n}\left\{ {{m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {m_{sawbn}\cos \quad \alpha} +} \right.}} \\{{\quad \left. {{m_{sn}z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {m_{aw1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}}} \right\}} +} \\{\quad {2{\overset{.}{z}}_{0}\left\{ {{{\overset{.}{z}}_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right.}} \\{\quad {{\left. \left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right. \right\} m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {\overset{.}{\alpha}m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} +}} \\{{{\quad \left. {{\overset{.}{\alpha}m_{sawbn}\cos \quad \alpha} - {\overset{.}{\beta}m_{sawcn}}} \right\}}\cos \quad \beta} -} \\{\quad {2\overset{.}{\beta}{{\overset{.}{z}}_{0}\left\lbrack \left\{ {{z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right. \right.}}} \\\left. {{\quad \left. {{m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\sin \quad \alpha}} \right\}}\sin \quad \beta} \middle| - \right. \\{\quad {{g\left\{ {{z_{0}m_{b}} - {m_{ba}\sin \quad \beta} + {{m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}\cos \quad \beta}} \right\}} -}} \\{\quad {{\frac{1}{2}k_{zi}e_{0i}^{2}\left\{ {{\sin \left( {\gamma_{i} + \theta_{i}} \right)} + {\sin \left( {\gamma_{ii} + \theta_{ii}} \right)}} \right\}^{2}} -}} \\{\quad {{\frac{1}{2}k_{ziii}e_{0{iii}}^{2}\left\{ {{\sin \left( {\gamma_{iii} + \theta_{iii}} \right)} + {\sin \left( {\gamma_{iv} + \theta_{iv}} \right)}} \right\}^{2}} -}} \\{\quad {\sum\limits_{n = i}^{iv}\quad {\langle{g\left\lbrack {{z_{0}m_{sawn}} + \left\{ {{m_{sn}z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right.} \right.}}}} \\{\quad {{m_{aw1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} +}} \\{\left. {{{\quad \left. {m_{sawbn}\sin \quad \alpha} \right\}}\cos \quad \beta} - {m_{sawan}\sin \quad \beta}} \right\rbrack + {\frac{1}{2}{k_{sn}\left( {z_{6n} - l_{sn}} \right)}^{2}} +} \\{\quad {{\frac{1}{2}{k_{wn}\left( {z_{12n} - l_{wn}} \right)}^{2}}\rangle}}\end{matrix} & (65) \\{\frac{\partial L}{\partial z_{0}} = {- {g\left( {m_{b} + m_{sawn}} \right)}}} & \quad \\\begin{matrix}{\frac{\partial L}{\partial{\overset{.}{z}}_{0}} = \quad {{{\overset{.}{z}}_{0}m_{b}} + {\overset{.}{\alpha}m_{b}\cos \quad {\beta \left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} - {\beta \left\{ {{m_{ba}\cos \quad \beta} +} \right.}}} \\{{\quad \left. {{m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}\sin \quad \beta} \right\}} + {{\overset{.}{z}}_{0}m_{sawn}} +} \\{\quad \left\{ {{z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right.} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {\overset{.}{\alpha}m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} +}} \\{{{\quad \left. {{\overset{.}{\alpha}m_{sawbn}\cos \quad \alpha} - {\overset{.}{\beta}m_{sawan}}} \right\}}\cos \quad \beta} -} \\{\quad {\overset{.}{\beta}\left\{ {{m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right.}} \\{{\quad \left. {{m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\sin \quad \alpha}} \right\}}\sin \quad \beta}\end{matrix} & \quad \\\begin{matrix}{{\frac{\quad}{t}\left( \frac{\partial L}{\partial{\overset{.}{z}}_{0}} \right)} = \quad {{{\overset{¨}{z}}_{0}\left( {m_{b} + m_{sawn}} \right)} + {\overset{¨}{\alpha}{m_{b}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} -}} \\{\quad {{\overset{.}{\beta}\overset{.}{\alpha}m_{b}\sin \quad {\beta \left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} +}} \\{\quad {{{\overset{.}{\alpha}}^{2}m_{b}\cos \quad {\beta \left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}} -}} \\{\quad {{\overset{¨}{\beta}\left\{ {{m_{ba}\cos \quad \beta} + {{m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}\sin \quad \beta}} \right\}} +}} \\{\quad {\overset{.}{\beta}\left\{ {{\overset{.}{\beta}m_{ba}\sin \quad \beta} + {\overset{.}{\alpha}{m_{b}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}\sin \quad \beta} +} \right.}} \\{{\quad \left. {\overset{.}{\beta}{m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}\cos \quad \beta} \right\}} +} \\{\quad \left\{ {{{\overset{¨}{z}}_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -} \right.} \\{{\quad \left. {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right){\overset{.}{z}}_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} \right\}} -} \\{\quad {{\left( {\overset{¨}{\alpha} + {\overset{¨}{\eta}}_{n}} \right)z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right){\overset{.}{z}}_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)^{2}z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{\quad {{\overset{¨}{\alpha}m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {{\overset{.}{\alpha}}^{2}m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} +}} \\{{{\quad \left. {{\overset{¨}{\alpha}m_{sawbn}\cos \quad \alpha} - {{\overset{.}{\alpha}}^{2}m_{sawbn}\sin \quad \alpha} - {\overset{¨}{\beta}m_{sawan}}} \right\}}\cos \quad \beta} -} \\{\quad {\overset{.}{\beta}\left\{ {{{\overset{.}{z}}_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -} \right.}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{\quad {{\overset{.}{\alpha}m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {\overset{.}{\alpha}m_{sawbn}\cos \quad \alpha} -}} \\{{{\quad \left. {\overset{.}{\beta}m_{sawan}} \right\}}\sin \quad \beta} - {\overset{¨}{\beta}\left\{ {{m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right.}} \\{\quad {{z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {m_{awcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} +}} \\{{{\quad \left. {m_{sawbn}\sin \quad \alpha} \right\}}\sin \quad \beta} -} \\{\quad {\overset{.}{\beta}\left\{ {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right.}} \\{\quad {{z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{{{\quad \left. {{\overset{.}{\alpha}m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {\overset{.}{\alpha}m_{sawbn}\cos \quad \alpha}} \right\}}\sin \quad \beta} -} \\{\quad {{\overset{.}{\beta}}^{2}\left\{ {{m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -} \right.}} \\{\quad {{z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} +}} \\{{\quad \left. {m_{sawbn}\sin \quad \alpha} \right\}}\cos \quad \beta}\end{matrix} & \quad \\\begin{matrix}{\frac{\partial L}{\partial\beta} = \quad {{{- \overset{.}{\alpha}}{\overset{.}{z}}_{0}m_{b}\sin \quad {\beta \left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} +}} \\{\quad \left. {\overset{.}{\beta}{\overset{.}{z}}_{0}\left\{ {{m_{ba}\sin \quad \beta} - {{m_{b}\left( {{b_{0}\sin \quad \alpha} - {c_{0}\cos \quad \alpha}} \right)}\cos \quad \beta}} \right\}} \right)} \\{\quad {{g\left\{ {{m_{ba}\cos \quad \beta} + {{m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}\sin \quad \beta}} \right\}} +}} \\{\quad {\langle{g\left\lbrack \left\{ {{m_{sn}z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right. \right.}}} \\{\left. {{{\quad \left. {{m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\sin \quad \alpha}} \right\}}\sin \quad \beta} + {m_{sawan}\cos \quad \beta}} \right\rbrack -} \\{\quad {{\overset{.}{z}}_{0}\left\{ {{{\overset{.}{z}}_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right.}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {\overset{.}{\alpha}m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} +}} \\{{{\quad \left. {{\overset{.}{\alpha}m_{sawbn}\cos \quad \alpha} - {\overset{.}{\beta}m_{sawan}}} \right\}}\sin \quad \beta} +} \\{\quad {\overset{.}{\beta}{\overset{.}{z}}_{0}\left\{ {{m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right.}} \\{{{\quad \left. {{m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\sin \quad \alpha}} \right\}}\cos \quad \beta}\rangle}\end{matrix} & (66) \\\begin{matrix}{\frac{\partial L}{\partial\alpha} = \quad {{\left\{ {{{\overset{.}{\beta}}^{2}{m_{b}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} + {\overset{.}{\alpha}\overset{.}{\beta}m_{ba}}} \right\} \left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)} -}} \\{\quad {{\overset{.}{\alpha}{\overset{.}{z}}_{0}m_{b}\cos \quad {\beta \left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}} -}} \\{\quad {{\overset{.}{\beta}{\overset{.}{z}}_{0}{m_{b}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}\sin \quad \beta} +}} \\{\quad \left| {{\overset{.}{\beta}}^{2}{\langle{m_{sn}\left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}}} \right.} \\{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\left\{ {{{- z_{6n}}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {c_{1n}{\sin \left( {\alpha + \gamma_{n}} \right)}} +} \right.} \\{{\quad \left. {b_{2n}\cos \quad \alpha} \right\}} + {m_{an}\left\{ {{e_{1\quad n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\left\{ {{e_{1}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} +} \right.} \\{{\quad \left. {b_{2n}\cos \quad \alpha} \right\}} + {m_{wn}\left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\left\{ {{e_{3}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} +} \right.} \\{{{\quad \left. {b_{2n}\cos} \right\}}\rangle} + {{\overset{.}{z}}_{6n}\overset{.}{\beta}m_{sn}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \\{\quad {{{\overset{.}{\eta}}_{n}\overset{.}{\beta}m_{sn}z_{6n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{\overset{.}{\theta}\overset{.}{\beta}m_{aw1n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +}} \\{\quad {\overset{.}{\alpha}\overset{.}{\beta}a_{1n}\left\{ {{m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\sin \quad \alpha} +} \right.}} \\{{\quad \left. {{m_{sn}z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {m_{aw1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}}} \right\}} -} \\{\quad {{\overset{.}{z}}_{0}\left( {{{\overset{.}{z}}_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right.}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{aw1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {\overset{.}{\alpha}m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} +}} \\{{{\quad \left. {\overset{.}{\alpha}m_{sawbn}\sin \quad \alpha} \right\}}\cos \quad \beta} - {\overset{.}{\beta}{{\overset{.}{z}}_{0}\left\lbrack \left\{ {{m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -} \right. \right.}}} \\{\quad {{z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} +}} \\\left. {{\quad \left. {m_{sawbn}\cos \quad \alpha} \right\}}\sin \quad \beta} \middle| {{{- {{gm}_{b}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}}\cos \quad \beta} +} \right. \\{\quad {g\left\{ {{m_{sn}z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right.}} \\{{\quad \left. {{m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {m_{sawbn}\cos \quad \alpha}} \right\}}\cos \quad \beta}\end{matrix} & (67) \\\begin{matrix}{\frac{\partial L}{\partial\eta_{n}} = \quad {{{\overset{.}{\alpha}}^{2}m_{sn}z_{6n}\left\{ {{{- c_{1n}}\sin \quad \eta_{n}} - {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{\overset{.}{\beta}}^{2}m_{sn}\left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\left\{ {{- z_{6n}}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} \right\}} +} \\{\quad {{{\overset{.}{z}}_{6n}\overset{.}{\alpha}m_{sn}\left\{ {{c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{{\overset{.}{z}}_{6n}{\overset{.}{\beta}}_{sn}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{\quad {{{\overset{.}{\eta}}_{n}\overset{.}{\alpha}m_{sn}z_{6n}\left\{ {{c_{1n}\sin \quad \eta_{n}} + \quad {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{{\overset{.}{\eta}}_{n}\overset{.}{\beta}m_{sn}z_{6n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{\overset{.}{\alpha}\overset{.}{\beta}a_{1n}m_{sn}z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{{gm}_{sn}z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} -}} \\{\quad {{\overset{.}{z}}_{0}\left\{ {{{\overset{.}{z}}_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right.}} \\{{{\quad \left. {\left( {\overset{.}{\alpha} + \eta_{n}} \right)z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} \right\}}\cos \quad \beta} +} \\{\quad {\overset{.}{\beta}{\overset{.}{z}}_{0}z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\sin \quad \beta}}\end{matrix} & (68) \\\begin{matrix}{\frac{\partial L}{\partial\theta_{n}} = \quad {{- k_{zi}}e_{0i}^{2}\left\{ {{\sin \left( {\gamma_{i} + \theta_{i}} \right)} + {\sin \left( {\gamma_{ii} + \theta_{ii}} \right)}} \right\} \left\{ {{\cos \left( {\gamma_{i} + \theta_{i}} \right)} +} \right.}} \\{{\quad \left. {\cos \left( {\gamma_{ii} + \theta_{ii}} \right)} \right\}} - {k_{ziii}e_{0{iii}}^{2}\left\{ {{\sin \left( {\gamma_{iii} + \theta_{iii}} \right)} +} \right.}} \\{{{\quad \left. {\sin \left( {\gamma_{iv} + \theta_{iv}} \right)} \right\}}\left\{ {{\cos \left( {\gamma_{iii} + \theta_{iii}} \right)} + {\cos \left( {\gamma_{iv} + \theta_{iv}} \right)}} \right\}} -} \\{\quad {{{\overset{.}{\alpha}}^{2}m_{aw1n}\left\{ {{c_{2n}\cos \quad \theta_{n}} + {b_{2n}{\sin \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}} +}} \\{\quad {{\overset{.}{\beta}}^{2}{\langle{m_{an}\left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}}}} \\{{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}e_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \\{\quad {m_{wn}\left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}e_{3n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}}\rangle} -} \\{\quad {{\overset{.}{\theta}\overset{.}{\alpha}m_{aw1n}\left\{ {{c_{2n}\cos \quad \theta_{n}} + {b_{2n}{\sin \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}} +}} \\{\quad {{\overset{.}{\theta}\overset{.}{\beta}m_{aw1n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +}} \\{\quad {{\overset{.}{\alpha}\overset{.}{\beta}a_{1n}m_{aw1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -}} \\{\quad {{{gm}_{aw1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\cos \quad \beta} -}} \\{\quad {{{{\overset{.}{z}}_{0}\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)}m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\cos \quad \beta} -}} \\{\quad {\overset{.}{\beta}{\overset{.}{z}}_{0}m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\sin \quad \beta}}\end{matrix} & (69) \\\begin{matrix}{\frac{\partial L}{\partial z_{6n}} = \quad {{m_{sn}{\overset{.}{\eta}}_{n}^{2}z_{6n}} + {{\overset{.}{\alpha}}^{2}{m_{sn}\left\lbrack {z_{6n} + \left\{ {{c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} \right\rbrack}} +}} \\{\quad {{\overset{.}{\beta}}^{2}m_{sn}\left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \\{\quad {{{\overset{.}{\eta}}_{n}\overset{.}{\alpha}m_{sn}\left\{ {{2z_{6n}} + {c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{{\overset{.}{\eta}}_{n}\overset{.}{\beta}m_{sn}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{\overset{.}{\alpha}\overset{.}{\beta}a_{1n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{\quad {{{gm}_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} - {k_{sn}\left( {z_{6n} - l_{sn}} \right)} -}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right){\overset{.}{z}}_{0}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} -}} \\{\quad {\overset{.}{\beta}{\overset{.}{z}}_{0}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\sin \quad \beta}}\end{matrix} & (70) \\{\frac{\partial L}{\partial z_{12n}} = {- {k_{wn}\left( {z_{12n} - l_{wn}} \right)}}} & (71) \\\begin{matrix}{\frac{\partial L}{\partial\overset{.}{\beta}} = \quad {\overset{.}{\beta}{\langle{m_{saw2n} + m_{bal} + {m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}^{2} +}}}} \\{\quad {{m_{sn}\left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}^{2}} +}} \\{\quad {{m_{an}\left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + y_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}^{2}} +}} \\{{\quad {{m_{wn}\left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}^{2}}\rangle}} -} \\{\quad {{\overset{.}{\alpha}{m_{ba}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} - {{\overset{.}{z}}_{6}m_{sn}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{{\overset{.}{\eta}}_{n}m_{sn}z_{6n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{\quad {{\overset{.}{\theta}m_{aw1n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {\overset{.}{\alpha}a_{1n}\left\{ {{m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} -} \right.}}} \\{\quad {{m_{sawbn}\cos \quad \alpha} + {m_{sn}z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{{\quad \left. {m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} \right\}} - {{\overset{.}{z}}_{0}\left\lbrack {\left. \left\{ {{m_{b}b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right. \right) +} \right.}} \\{\quad {{m_{awln}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{{{\quad \left. {{m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\sin \quad \alpha}} \right\}}\sin \quad \beta} +} \\{\quad \left. {\left( {m_{ba} + m_{sawcn}} \right)\cos \quad \beta} \right\rbrack}\end{matrix} & (72) \\\begin{matrix}{{\frac{\quad}{t}\left( \frac{\partial L}{\partial\overset{.}{\beta}} \right)} = \quad {\overset{¨}{\beta}{\langle{m_{saw2n} + m_{bal} + {m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}^{2} +}}}} \\{\quad {m_{sn}\left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}^{2} + {m_{an}\left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right.}} \\{\quad {\left( \left. {c_{2n}\left( {{\cos \left( {\alpha + \gamma_{n}} \right)} + {b_{2n}\sin \quad \alpha}} \right.} \right\} \right)^{2} +}} \\{\quad {m_{wn}\left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}^{2}\rangle} + {2\overset{.}{\beta}{\langle{\overset{.}{\alpha}{m_{b}\left( {{b_{0}\sin \quad \alpha} +} \right.}}}}} \\{{{\quad \left. {c_{0}\cos \quad \alpha} \right)}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)} +} \\{\quad {m_{sn}\left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\left\{ {{{\overset{.}{z}}_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -} \right.} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {\overset{.}{\alpha}\left\lbrack {{c_{1n}{\sin \left( {\alpha + \gamma_{n}} \right)}} -} \right.}}} \\{\left. {\quad \left. {b_{2n}\cos \quad \alpha} \right\rbrack} \right\} + {m_{an}\left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right.}} \\{\quad {{c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +}} \\{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\left\{ {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -} \right.} \\{{\quad \left. {\overset{.}{\alpha}\left\lbrack {{c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {b_{2n}\cos \quad \alpha}} \right\rbrack} \right\}} +} \\{\quad {m_{wn}\left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\left\{ {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -} \right.} \\{{{\quad \left. {\overset{.}{\alpha}\left\lbrack {{c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {b_{2n}\cos \quad \alpha}} \right\rbrack} \right\}}\rangle} -} \\{\quad {{\overset{¨}{\alpha}{m_{ba}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} +}} \\{\quad {{{\overset{.}{\alpha}}^{2}{m_{ba}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}} -}} \\{\quad {{{\overset{¨}{z}}_{6n}m_{sn}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{{{\overset{.}{z}}_{6n}\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)}m_{sn}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{{\overset{¨}{\eta}}_{n}m_{sn}z_{6n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{{\overset{.}{\eta}}_{n}m_{sn}{\overset{.}{z}}_{6n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{{{\overset{.}{\eta}}_{n}\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)}m_{sn}z_{6n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{\quad {{{\overset{¨}{\theta}}_{n}m_{aw1n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +}} \\{\quad {{{{\overset{.}{\theta}}_{n}\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)}m_{aw1n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +}} \\{\quad {\overset{¨}{\alpha}a_{1n}\left\{ {{m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {m_{sawbn}\cos \quad \alpha} +} \right.}} \\{{\quad \left. {{m_{sn}z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {m_{aw1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}}} \right\}} +} \\{\quad {\overset{.}{\alpha}a_{1n}\left\{ {{\overset{.}{\alpha}m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {\overset{.}{\alpha}m_{sawbn}\sin \quad \alpha} +} \right.}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)m_{sn}z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{m_{sn}{\overset{.}{z}}_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{{\quad \left. {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{aw1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} \right\}} -} \\{\quad {{\overset{¨}{z}}_{0}\left\lbrack \left\{ {{m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)} + {m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right. \right.}} \\{\quad {{z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} +}} \\{{{\quad \left. {m_{sawbn}\sin \quad \alpha} \right\}}\sin \quad \beta} + \left. \left( {m_{ba} + {m_{sawan}\cos \quad \beta}} \right. \right\rbrack -} \\{\quad {{\overset{.}{z}}_{0}\left\lbrack \left\{ {{\overset{.}{\alpha}{m_{b}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} +} \right. \right.}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +}} \\{\quad {{{\overset{.}{z}}_{6n}{m_{sn}\left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{{{\quad \left. {{\overset{.}{\alpha}m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {\overset{.}{\alpha}m_{sawbn}\cos \quad \alpha}} \right\}}\sin \quad \beta} +} \\{\quad {\overset{.}{\beta}\left\{ {{m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)} + {m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right.}} \\{\quad {{z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} +}} \\\left. {{{\quad \left. {m_{sawbn}\sin \quad \alpha} \right\}}\cos \quad \beta} - {{\overset{.}{\beta}\left( {m_{ba} + m_{sawan}} \right)}\sin \quad \beta}} \right\rbrack\end{matrix} & (73) \\\begin{matrix}{\frac{\partial L}{\partial\overset{.}{\alpha}} = \quad {{\overset{.}{\alpha}m_{b\quad b\quad l}} - {\overset{.}{\beta}{m_{ba}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} +}} \\{\quad {{{\overset{.}{z}}_{0}m_{b}\cos \quad {\beta \left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} + {\overset{.}{\alpha}{\langle{m_{saw1n} +}}}}} \\{\quad {{m_{sn}{z_{6n}\left\lbrack {z_{6n} + {2\left\{ {{c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}}} \right\rbrack}} -}} \\{{\quad {{2m_{aw1n}\left\{ {{c_{2n}\sin \quad \theta_{n}} - {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}}\rangle}} +} \\{\quad {{{\overset{.}{z}}_{6n}m_{sn}\left\{ {{c_{1n}\sin \quad \eta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{{\overset{.}{\eta}}_{n}m_{sn}z_{6n}\left\{ {z_{6n} + {c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{\overset{.}{\theta}\left\lbrack {m_{aw21n} - {m_{aw1n}\left\{ {{c_{2n}\sin \quad \theta_{n}} - {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}}} \right\rbrack} +}} \\{\quad {\overset{.}{\beta}a_{1n}\left\{ {{m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {m_{sawbn}\cos \quad \alpha} +} \right.}} \\{{\quad \left. {{m_{sn}z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {m_{aw1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}}} \right\}} +} \\{\quad {{\overset{.}{z}}_{0}\left\{ {{m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -} \right.}} \\{{\quad \left. {{m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\cos \quad \alpha}} \right\}}\cos \quad \beta}\end{matrix} & (74) \\\begin{matrix}{{\frac{\quad}{t}\left( \frac{\partial L}{\partial\overset{.}{\alpha}} \right)} = \quad {{{- \overset{.}{\beta}}{m_{ba}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} +}} \\{\quad {{\overset{.}{\beta}\overset{.}{\alpha}{m_{ba}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}} +}} \\{\quad {{{\overset{¨}{z}}_{0}m_{b}\cos \quad {\beta \left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} -}} \\{\quad {{\overset{.}{\beta}{\overset{.}{z}}_{0}m_{b}\sin \quad {\beta \left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} -}} \\{\quad {{\overset{.}{\alpha}{\overset{.}{z}}_{0}m_{b}\cos \quad {\beta \left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}} + {\overset{¨}{\alpha}{\langle{m_{b\quad b\quad l} + m_{saw1n} +}}}}} \\{\quad {{m_{sn}{z_{6n}\left\lbrack {z_{6n} + {2\left\{ {{c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}}} \right\rbrack}} -}} \\{{\quad {{2m_{aw1n}\left\{ {{c_{2n}\sin \quad \theta_{n}} - {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}}\rangle}} +} \\{\quad {\overset{.}{\alpha}{\langle{{m_{sn}{{\overset{.}{z}}_{6n}\left\lbrack {z_{6n} + {2\left\{ {{c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}}} \right\rbrack}} +}}}} \\{\quad {{m_{sn}{z_{6n}\left\lbrack {{\overset{.}{z}}_{6n} - {2{\overset{.}{\eta}}_{n}\left\{ {{c_{1n}\sin \quad \eta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}}} \right\rbrack}} -}} \\{{\quad {{2{\overset{.}{\theta}}_{n}m_{aw1n}\left\{ {{c_{2n}\cos \quad \theta_{n}} + {b_{2n}{\sin \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}}\rangle}} +} \\{\quad {{{\overset{¨}{z}}_{6n}m_{sn}\left\{ {{c_{1n}\sin \quad \eta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{{\overset{.}{z}}_{6n}{\overset{.}{\eta}}_{n}m_{sn}\left\{ {{c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{{\overset{¨}{\eta}}_{n}m_{sn}z_{6n}\left\{ {z_{6n} + {c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{{\overset{.}{\eta}}_{n}m_{sn}{\overset{.}{z}}_{6n}\left\{ {z_{6n} + {c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{{\overset{.}{\eta}}_{n}m_{sn}z_{6n}\left\{ {{\overset{.}{z}}_{6n} - {{\overset{.}{\eta}}_{n}\left\lbrack {{c_{1n}\sin \quad \eta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\rbrack}} \right\}} +}} \\{\quad {{\overset{¨}{\theta}\left\lbrack {m_{aw2ln} - {m_{aw1n}\left\{ {{c_{2n}\sin \quad \theta_{n}} - {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}}} \right\rbrack} -}} \\{{\quad \left. {{\overset{.}{\theta}}_{n}^{2}m_{aw1n}\left\{ {{c_{2n}\cos \quad \theta_{n}} + {b_{2n}{\sin \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}} \right\rbrack} +} \\{\quad {\overset{¨}{\beta}a_{1n}\left\{ {{m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {m_{sawbn}\cos \quad \alpha} +} \right.}} \\{{\quad \left. {{m_{sn}z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {m_{aw1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}}} \right\}} +} \\{\quad {\overset{.}{\beta}a_{1n}\left\{ {{\overset{.}{\alpha}\left\lbrack {{m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\sin \quad \alpha}} \right\rbrack} +} \right.}} \\{\quad {{m_{sn}{\overset{.}{z}}_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)m_{sn}z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{{\quad \left. {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{aw1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} \right\}} -} \\{\quad {{\overset{¨}{z}}_{0}\left\{ {{m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right.}} \\{\quad {{z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} +}} \\{{{\quad \left. {m_{sawbn}\cos \quad \alpha} \right\}}\cos \quad \beta} -} \\{\quad {{\overset{.}{z}}_{0}\left\{ {{{- \left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)}m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right.}} \\{\quad {{{\overset{.}{z}}_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{{{\quad \left. {{\overset{.}{\alpha}m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} - {\overset{.}{\alpha}m_{sawbn}\sin \quad \alpha}} \right\}}\cos \quad \beta} -} \\{\quad {\overset{.}{\beta}{\overset{.}{z}}_{0}\left\{ {{m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -} \right.}} \\{\quad {{z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{{\quad \left. {{m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\cos \quad \alpha}} \right\}}\sin \quad \beta}\end{matrix} & (75) \\\begin{matrix}{\frac{\partial L}{\partial{\overset{.}{\eta}}_{n}} = \quad {{m_{sn}{\overset{.}{\eta}}_{n}z_{6n}^{2}} + {\overset{.}{\alpha}m_{sn}z_{6n}\left\{ {z_{6n} + {c_{1n}\cos \quad \eta_{n}} -} \right.}}} \\{{\quad \left. {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}} \right\}} + {\overset{.}{\beta}m_{sn}z_{6n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -} \\{\quad {{\overset{.}{z}}_{0}z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta}}\end{matrix} & (76) \\\begin{matrix}{{\frac{\quad}{t}\left( \frac{\partial L}{\partial{\overset{.}{\eta}}_{n}} \right)} = \quad {{m_{sn}{\overset{¨}{\eta}}_{n}z_{6n}^{2}} + {2m_{sn}{\overset{.}{\eta}}_{n}{\overset{.}{z}}_{6n}z_{6n}} +}} \\{\quad {{\overset{¨}{\alpha}m_{sn}z_{6n}\left\{ {z_{6n} + {c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{\overset{.}{\alpha}m_{sn}{\overset{.}{z}}_{6n}\left\{ {z_{6n} + {c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{\overset{.}{\alpha}m_{sn}z_{6n}\left\{ {{\overset{.}{z}}_{6n} - {{\overset{.}{\eta}}_{n}\left\lbrack {{c_{1n}\sin \quad \eta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\rbrack}} \right\}} +}} \\{\quad {{\overset{¨}{\beta}m_{sn}z_{6n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{\overset{.}{\beta}m_{sn}{\overset{.}{z}}_{6n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{{\overset{.}{\beta}\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)}m_{sn}z_{6n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{\quad {{{\overset{¨}{z}}_{0}z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} -}} \\{\quad {{{\overset{.}{z}}_{0}z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} -}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right){\overset{.}{z}}_{0}z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} -}} \\{\quad {\overset{.}{\beta}{\overset{.}{z}}_{0}z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta}}\end{matrix} & {(77)\quad} \\\begin{matrix}{\frac{\partial L}{\partial{\overset{.}{\theta}}_{n}} = \quad {{{\overset{.}{\theta}}_{n}m_{aw2ln}} + {\overset{.}{\alpha}\left\lbrack {m_{aw21n} - {m_{aw1n}\left\{ {{c_{2n}\sin \quad \theta_{n}} - {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}}} \right\rbrack} -}} \\{\quad {{\overset{.}{\beta}m_{aw1n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {{\overset{.}{z}}_{0}m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\cos \quad \beta}}}\end{matrix} & (78) \\\begin{matrix}{{\frac{\quad}{t}\left( \frac{\partial L}{\partial{\overset{.}{\theta}}_{n}} \right)} = \quad {{{\overset{¨}{\theta}}_{n}m_{aw21n}} + {\overset{¨}{\alpha}\left\lbrack {m_{aw2ln} - {m_{aw1n}\left\{ {{c_{2n}\sin \quad \theta_{n}} -} \right.}} \right.}}} \\{\quad {\overset{.}{\left. \left. {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}} \right\} \right\rbrack - \alpha}{\overset{.}{\theta}}_{n}m_{awln}\left\{ {{c_{2n}\cos \quad \theta_{n}} +} \right.}} \\{\quad {{\overset{¨}{\left. {b_{2n}{\sin \left( {\gamma_{n} + \theta_{n}} \right)}} \right\} - \beta}m_{aw1n}\alpha_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +}} \\{\quad {{{\overset{.}{\beta}\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)}m_{aw1n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +}} \\{\quad {{{\overset{¨}{z}}_{0}m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\cos \quad \beta} -}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right){\overset{.}{z}}_{0}m_{aw1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\cos \quad \beta} -}} \\{\quad {\overset{.}{\beta}{\overset{.}{z}}_{0}m_{aw1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\sin \quad \beta}}\end{matrix} & (79) \\\begin{matrix}{\frac{\partial L}{\partial{\overset{.}{z}}_{6n}} = \quad {{m_{sn}{\overset{.}{z}}_{6n}} + {\overset{.}{\alpha}m_{sn}\left\{ {{c_{1n}\sin \quad \eta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} -}} \\{\quad {{\overset{.}{\beta}m_{sn}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {{\overset{.}{z}}_{0}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta}}}\end{matrix} & (80) \\\begin{matrix}{{\frac{\quad}{t}\left( \frac{\partial L}{\partial{\overset{.}{z}}_{6n}} \right)} = \quad {{m_{sn}{\overset{¨}{z}}_{6n}} + {\overset{¨}{\alpha}m_{sn}\left\{ {{c_{1n}\sin \quad \eta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{\overset{.}{\alpha}{\overset{.}{\eta}}_{n}m_{sn}\left\{ {{c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} -}} \\{\quad {{\overset{¨}{\beta}m_{sn}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{{\overset{.}{\beta}\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)}m_{sn}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{{\overset{¨}{z}}_{0}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} -}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right){\overset{.}{z}}_{0}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} -}} \\{\quad {\overset{.}{\beta}{\overset{.}{z}}_{0}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\sin \quad \beta}}\end{matrix} & (81) \\{\frac{\partial L}{\partial{\overset{.}{z}}_{12n}} = 0} & (82) \\{{\frac{\quad}{t}\left( \frac{\partial L}{\partial{\overset{.}{z}}_{12n}} \right)} = 0} & (83)\end{matrix}$

The dissipative function is:

F_(tot)=−1/2(c_(sn){dot over (z)}_(6n) ²+c_(wn){dot over (z)}_(12n)²)  (84)

The constraints are based on geometrical constraints, and the touchpoint of the road and the wheel. The geometrical constraint is expressedas

e_(2n) cos θ_(n)=−(z_(6n)−d_(1n))sin η_(n)

e_(2n) sin θ_(n)−(z_(6n)−d_(1n))cos η_(n)=c_(1n)−c_(2n)  (85)

The touch point of the road and the wheel is defined as $\begin{matrix}\begin{matrix}{z_{tn} = \quad z_{P_{{touchpoint},n}^{r}}} \\{\quad {z_{0} + \left\{ {{z_{12n}\cos \quad \alpha} + {e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right.}} \\{{{\quad \left. {{c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}}\cos \quad \beta} - {a_{1n}\sin \quad \beta}} \\{= \quad {R_{n}(t)}}\end{matrix} & (86)\end{matrix}$

where R_(n)(t) is road input at each wheel.

Differentials are:

{dot over (θ)}_(n)e_(2n) sin θ_(n)−{dot over (z)}_(6n) sin {dot over(η)}_(n)(z_(6n)−d_(1n))cos η_(n)=0

{dot over (θ)}_(n)e_(2n) cos θ_(n)−{dot over (z)}_(6n) cos η_(n)+{dotover (η)}_(n)(z_(6n)−d_(1n))sin η_(n)=0

{dot over (z)}₀+{{dot over (z)}_(12n) cos α−{dot over (α)}z_(12n) sin{dot over (α)}+{dot over (θ)}_(n))e_(3n) cos(α+γ_(n)+θ_(n)) −{dot over(α)}c_(2n) sin(α+γ_(n))+{dot over (α)}b_(2n) cos α}cos β−{dot over(β)}[{z_(12n) cos α+e_(3n) sin(α+γ_(n)+θ_(n)) +c_(2n)cos(α+γ_(n))+b_(2n) sin α}sin β+a_(1n) cos β]−{dot over(R)}_(n)(t)=0  (87)

Since the differentials of these constraints are written as$\begin{matrix}{{{\sum\limits_{j}{a_{lnj}d{\overset{.}{q}}_{j}}} + {a_{lnt}{t}}} = {0\quad \left( {{l = 1},2,{{3\quad n} = i},{ii},{iii},{iv}} \right)}} & (88)\end{matrix}$

then the values a_(lnj) are obtained as follows.

a_(1n0)=0

a_(2n0)=0

a_(3n0)=1

a_(1n1)=0, a_(1n2)=0, a_(1n3)=−(z_(6n)−d_(1n))cos η_(n), a_(1n4)=e_(2n)sin θ_(n), a_(1n5)=−sin η_(n), a_(1n6)=0

a_(2n1)=0, a_(2n2)=0, a_(2n3)=(z_(6n)−d_(1n))sin η_(n), a_(2n4)=e_(2n)cos θ_(n), a_(2n5)=−cos η_(n), a_(2n6)=0

a_(3n1)=−{z₁₂ cos α+e_(3n) sin(α+γ_(n)+θ_(n))+b_(2n) cos(α+γ_(n))+b_(2n)sin α}sin β+a_(1n) cos β,

a_(3n2)={−z_(12n) sin α+e_(3n) cos(α+γ_(n)+θ_(n))−c_(2n)sin(α+γ_(n))+b_(2n) cos α}cos β,

a_(3n3)=0, a_(3n4)=e_(3n) cos(α+γ_(n)+θ_(n))cos β, a_(3n5)=0,a_(3n6)=cos α cos β  (89)

From the above, Lagrange's equation becomes $\begin{matrix}{{{\frac{}{t}\left( \frac{\partial L}{\partial{\overset{.}{q}}_{j}} \right)} - \frac{\partial L}{\partial q_{j}}} = {Q_{j} + {\sum\limits_{l,n}{\lambda_{l\quad n}a_{lnj}}}}} & (90)\end{matrix}$

where

q₀=z₀

q₁=β, q₂=α, q_(3i)=η_(i), q_(4i)=θ_(i), q_(5i)=z_(6i), q_(6i)=z_(12i)

q_(3ii)=η_(ii), q_(4ii)=θ_(ii), q_(5ii)=z_(6ii), q_(6ii)=z_(12ii),

q_(3iii)=η_(iii), q_(4iii)=θ_(iii), q_(5iii)=z_(6iii),q_(6iii)=z_(12iii)

q_(3iv)=η_(iv), q_(4iv)=θ_(iv), q_(5iv)=z_(6v), q_(6iv)=z_(12iv)  (91)

${{{\frac{}{t}\left( \frac{\partial L}{\partial{\overset{.}{z}}_{0}} \right)} - \frac{\partial L}{\partial z_{0}}} = {{\frac{\partial F}{\partial{\overset{.}{z}}_{0}} + {\sum\limits_{l,n}{\lambda_{l\quad n}a_{l\quad {n0}}\quad l}}} = 1}},2,{{3\quad n} = i},{ii},{iii},{iv}$${{{\overset{¨}{z}}_{0}\left( {m_{b} + m_{sawn}} \right)} + {\overset{¨}{\alpha}m_{b}\cos \quad {\beta \left( {{b_{0}\quad \cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} - {\overset{.}{\beta}\overset{.}{\alpha}m_{b}\sin \quad {\beta \left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} - {{\overset{.}{\alpha}}^{2}m_{b}\cos \quad {\beta \left( {{b_{0}\sin \quad \alpha} - {c_{0}\cos \quad \alpha}} \right)}} - {\overset{¨}{\beta}\left\{ {{m_{ba}\cos \quad \beta} + {{m_{b}\left( {{b_{0}\sin \quad \alpha} - {c_{0}\cos \quad \alpha}} \right)}\sin \quad \beta}} \right\}} + {\overset{.}{\beta}\left\{ {{\overset{.}{\beta}m_{ba}\sin \quad \beta} + {\overset{.}{\alpha}{m_{b}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}\sin \quad \beta} + {\overset{.}{\beta}{m_{b}\left( {{b_{0}\sin \quad \alpha} - {c_{0}\cos \quad \alpha}} \right)}\cos \quad \beta}} \right\}} + {\left\{ {{z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right){\overset{.}{z}}_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {\left( {\overset{¨}{\alpha} + {\overset{¨}{\theta}}_{n}} \right)m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)^{2}m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {\left( {\overset{¨}{\alpha} + {\overset{¨}{\theta}}_{n}} \right)z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)^{2}z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)^{2}z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {\overset{¨}{\alpha}m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {{\overset{.}{\alpha}}^{2}m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {\overset{¨}{\alpha}m_{sawbn}\cos \quad \alpha} - {{\overset{.}{\alpha}}^{2}m_{sawbn}\sin \quad \alpha} - {\overset{¨}{\beta}m_{sawan}}} \right\} \cos \quad \beta} - {\overset{.}{\beta\{}{\overset{.}{z}}_{6n}m_{sn}\cos \quad \left( {\alpha + \gamma_{n} + \eta_{n}} \right)} + {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {\overset{.}{\alpha}m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {\overset{.}{\alpha}m_{sawbn}\cos \quad \alpha} - {\overset{.}{\beta}m_{sawan}\text{\}}\sin \quad \beta} - {\overset{¨}{\beta}\left\{ {{m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\sin \quad \alpha}} \right\} \sin \quad \beta} - {\overset{.}{\beta}\left\{ {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {{\overset{.}{z}}_{6n}m_{sn}\cos \quad \left( {\alpha + \gamma_{n} + \eta_{n}} \right)} + {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}m_{sn}\sin \quad \left( {\alpha + \gamma_{n} + \eta_{n}} \right)} - {\overset{.}{\alpha}m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {\overset{.}{\alpha}m_{sawbn}\cos \quad \alpha}} \right\} \sin \quad \beta} - {{\overset{.}{\beta}}^{2}\left\{ {{m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\sin \quad \alpha}} \right\} \cos \quad \beta} + {g\left( {m_{b} + m_{sawn}} \right)}} = {{\lambda_{3n}{{\overset{¨}{z}}_{0}\left( {m_{b} + m_{sawn}} \right)}} + {\overset{¨}{\alpha}m_{b}\cos \quad {\beta \left( {{b_{0}\quad \cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} - {{\overset{.}{\alpha}}^{2}m_{b}\cos \quad {\beta \left( {{b_{0}\sin \quad \alpha} - {c_{0}\cos \quad \alpha}} \right)}} - {\overset{¨}{\beta}\left\{ {{m_{ba}\cos \quad \beta} + {{m_{b}\left( {{b_{0}\sin \quad \alpha} - {c_{0}\cos \quad \alpha}} \right)}\sin \quad \beta}} \right\}} + {\overset{.}{\beta}\left\{ {{{\overset{.}{\beta}\left( {m_{ba} + m_{sawan}} \right)}\sin \quad \beta} + {\overset{¨}{\beta}{m_{b}\left( {{b_{0}\sin \quad \alpha} - {c_{0}\cos \quad \alpha}} \right)}\cos \quad \beta}} \right\}} + {\left\{ {{{\overset{¨}{z}}_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {2\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right){\overset{.}{z}}_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {\left( {\overset{¨}{\alpha} + {\overset{¨}{\theta}}_{n}} \right)m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)^{2}m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {\left( {\overset{¨}{\alpha} + {\overset{¨}{\theta}}_{n}} \right)z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)^{2}z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {\overset{¨}{\alpha}m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {{\overset{.}{\alpha}}^{2}m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {\overset{¨}{\alpha}m_{sawbn}\cos \quad \alpha} - {{\overset{.}{\alpha}}^{2}m_{sawbn}\sin \quad \alpha} - {\overset{¨}{\beta}m_{sawan}}} \right\} \cos \quad \beta} - {2\overset{.}{\beta}\left\{ {{{\overset{.}{z}}_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {\overset{.}{\alpha}m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {\overset{.}{\alpha}m_{sawbn}\cos \quad \alpha}} \right\} \sin \quad \beta} - {\left( {{\overset{¨}{\beta}\sin \quad \beta} + {{\overset{.}{\beta}}^{2}\cos \quad \beta}} \right)\left\{ {{{m_{awln}{\sin\left( {\left( {\alpha + \gamma_{n} + \theta_{n}} \right) + {z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\sin \quad \alpha}} \right\}}} + {g\left( {m_{b} + m_{sawn}} \right)}} = \lambda_{3n}} \right.}}$

$\begin{matrix}{{\overset{¨}{z}}_{0} = {\lambda_{3n} - g - \frac{\begin{matrix}{{\overset{¨}{\alpha}m_{b}C_{\beta}A_{2}} - {{\overset{.}{\alpha}}^{2}m_{b}A_{1}} - {\overset{¨}{\beta}\left\{ {{m_{bn}C_{\beta}} + {m_{b}A_{1}S_{\beta}}} \right\}} +} \\{{\overset{.}{\beta}\left\{ {{m_{ba}S_{\beta}} + {\overset{.}{\beta}m_{b}A_{1}C_{\beta}}} \right\}} + \left\{ {{{\overset{¨}{z}}_{6n}m_{sn}C_{\alpha \quad {\gamma\eta}}} - {2\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right){\overset{.}{z}}_{6n}m_{sn}S_{\alpha\gamma\eta}} +} \right.} \\{{\left( {\overset{¨}{\alpha} + {\overset{¨}{\theta}}_{n}} \right)m_{awln}C_{\alpha\gamma\eta}} - {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)^{2}m_{awln}S_{\alpha\gamma\eta}} - {\left( {\overset{¨}{\alpha} + {\overset{¨}{\eta}}_{n}} \right)z_{6n}m_{sn}S_{\alpha\gamma\eta}} -} \\{{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)^{2}z_{6n}m_{sn}C_{\alpha\gamma\eta}} - {\overset{¨}{\alpha}m_{sawcn}S_{\alpha\gamma\eta}} - {{\overset{.}{\alpha}}^{2}m_{sawcn}C_{\alpha\gamma\eta}} +} \\{{\left. {{\overset{¨}{\alpha}m_{sawcn}C_{\alpha}} - {{\overset{.}{\alpha}}^{2}m_{sawbn}S_{\alpha}} - {\overset{¨}{\beta}m_{sawan}}} \right\} C_{\beta}} - {2\overset{.}{\beta}\left\{ {{{\overset{.}{z}}_{6n}m_{sn}C_{\alpha\gamma\eta}} +} \right.}} \\{{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{awln}C_{\alpha\gamma\eta}} - {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}m_{sn}S_{\alpha\gamma\eta}} - {\overset{.}{\alpha}m_{sawcn}S_{\alpha\gamma\eta}} +} \\{{\left. {{\overset{.}{\alpha}m_{sawbn}C_{\alpha}} - {\overset{.}{\beta}{m_{sawan}/2}}} \right\} S\quad \beta} - {\left( {{\overset{¨}{\beta}S_{\beta}} + {{\overset{.}{\beta}}^{2}C_{\beta}}} \right)\left\{ {{m_{awln}S_{\alpha\gamma\eta}} +} \right.}} \\\left. {{z_{6n}m_{sn}C_{\alpha\gamma\eta}} + {m_{sawcn}C_{\alpha\gamma\eta}} + {m_{sawbn}S_{\alpha}}} \right\}\end{matrix}}{m_{bsawn}}}} & \quad \\{{{{\frac{\quad}{t}\left( \frac{\partial L}{\partial\overset{.}{\beta}} \right)} - \frac{\partial L}{\partial\beta}} = {{\frac{\partial F}{\partial\overset{.}{\beta}} + {\sum\limits_{l,n}{\lambda_{l\quad n}a_{{l\quad n},1}\quad l}}} = 1}},2,{{3\quad n} = i},{ii},{iii},{iv}} & (92) \\\begin{matrix}{\overset{¨}{\beta}{\langle\quad {m_{saw2n} + m_{bal} + {m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}^{2} +}}} \\{\quad {{m_{sn}\left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}^{2}} +}} \\{\quad {{m_{an}\left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}^{2}} +}} \\{{\quad {{m_{wn}\left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\}^{2}}\rangle}} +} \\{\quad {2\overset{.}{\beta}{\langle{{\overset{.}{\alpha}{m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)} +}}}} \\{\quad {m_{sn}\left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} \left\{ {{{\overset{.}{z}}_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -} \right.}} \\{{\quad \left. {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {\overset{.}{\alpha}\left\lbrack {{c_{1}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {b_{2n}\cos \quad \alpha}} \right\rbrack}} \right\}} +} \\{\quad {m_{an}\left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\left\{ {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {\overset{.}{\alpha}\left\lbrack {{c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {b_{2n}\cos \quad \alpha}} \right\rbrack}} \right\}} +} \\{\quad {m_{wn}\left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\left\{ {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {\overset{.}{\alpha}\left\lbrack {{c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {b_{2n}\cos \quad \alpha}} \right\rbrack}} \right\}}\rangle} -} \\{\quad {{\overset{¨}{\alpha}{m_{ba}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} + {{\overset{.}{\alpha}}^{2}{m_{ba}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}} -}} \\{\quad {{{\overset{¨}{z}}_{6n}m_{sn}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {{{\overset{.}{z}}_{6n}\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)}m_{sn}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{{\overset{¨}{\eta}}_{n}m_{sn}z_{6n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {{\overset{.}{\eta}}_{n}m_{sn}{\overset{.}{z}}_{6n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{{{\overset{.}{\eta}}_{n}\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)}m_{sn}z_{6n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {{\overset{¨}{\theta}}_{n}m_{aw1n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +}} \\{\quad {{{{\overset{.}{\theta}}_{n}\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)}m_{aw1n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {\overset{¨}{\alpha}a_{1n}\left\{ {{m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} -} \right.}}} \\{{\quad \left. {{m_{sawbn}\cos \quad \alpha} + {m_{sn}z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {m_{aw1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}}} \right\}} +} \\{\quad {\overset{.}{\alpha}a_{1n}\left\{ {{\overset{.}{\alpha}m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {\overset{.}{\alpha}m_{sawbn}\sin \quad \alpha} + {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)m_{sn}z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right.}} \\{{\quad \left. {{m_{sn}{\overset{.}{z}}_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{aw1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}}} \right\}} -} \\{\quad {{\overset{¨}{z}}_{0}\left\lbrack \left\{ {{m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)} + {m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right. \right.}} \\{\left. {{{\quad \left. {{m_{sawcn}{\cos \left( {\alpha + \gamma} \right)}} + {m_{sawbn}\sin \quad \alpha}} \right\}}\sin \quad \beta} + \left( {m_{ba} + {m_{sawan}\cos \quad \beta}} \right)} \right\rbrack -} \\{\quad {{\overset{.}{z}}_{0}\left\lbrack \left\{ {{\overset{.}{\alpha}{m_{b}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} + {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right. \right.}} \\{\quad {{{\overset{.}{z}}_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {\overset{.}{\alpha}m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} +}} \\{{{\quad \left. {\overset{.}{\alpha}m_{sawbn}\cos \quad \alpha} \right\}}\sin \quad \beta} + {\overset{.}{\beta}{\overset{.}{z}}_{0}\left\{ {{m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)} + {m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right.}} \\{{{\quad \left. {{z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\sin \quad \alpha}} \right\}}\cos \quad \beta} -} \\{{\quad \left. \left( {m_{ba} + {m_{sawan}\sin \quad \beta}} \right) \right\rbrack} + {\overset{.}{\alpha}{\overset{.}{z}}_{0}m_{b}\sin \quad {\beta \left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} -} \\{\quad {{\overset{.}{\beta}{\overset{.}{z}}_{0}\left\{ {{m_{ba}\sin \quad \beta} - {{m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}\cos \quad \beta}} \right\}} - {g\left\{ {{m_{ba}\cos \quad \beta} +} \right.}}} \\{{\quad \left. {{m_{b}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}\sin \quad \beta} \right\}} - {\langle{g\left\lbrack \left\{ {{m_{sn}z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right. \right.}}} \\{{{\quad \left. {{m_{aw1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\sin \quad \alpha}} \right\}}\sin \quad \beta} +} \\{{\quad \left. {m_{sawan}\cos \quad \beta} \right\rbrack} - {{\overset{.}{z}}_{0}\left\{ {{{\overset{.}{z}}_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -} \right.}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {\overset{.}{\alpha}m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {\overset{.}{\alpha}m_{sawbn}\cos \quad \alpha} -}} \\{\left. {{\quad \left. {\overset{.}{\beta}m_{sawan}} \right\}}\sin \quad \beta} \right\} - {\overset{.}{\beta}{\overset{.}{z}}_{0}\left\{ {{m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right.}} \\{{{\quad \left. {{m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\sin \quad \alpha}} \right\}}\cos \quad \beta}\rangle} \\{= \quad {\lambda_{3n}\left\lbrack {- \left\{ {{z_{12n}\cos \quad \alpha} + {e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.} \right.}} \\\left. {{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\sin \quad \beta} + {a_{1n}\cos \quad \beta}} \right\rbrack\end{matrix} & (93) \\\begin{matrix}{{\overset{¨}{\beta}\quad\left( {m_{saw2n} + m_{bal} + {m_{b}A_{1}^{2}} + {m_{sn}B_{1}^{2}} + {m_{an}B_{2}^{2}} + {m_{wn}B_{3}^{2}}} \right)} +} \\{\quad {2{\overset{.}{\beta}\left\lbrack {{\overset{.}{\alpha}m_{b}A_{1}A_{2}} + {m_{sn}B_{1}\left\{ {{{\overset{.}{z}}_{6n}C_{{\alpha\gamma\eta}\quad n}} - {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}S_{{\alpha\gamma\eta}\quad n}} - {\overset{.}{\alpha}A_{4}}} \right\}} +} \right.}}} \\{{\quad \left. {{m_{an}B_{2}\left\{ {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{1n}C_{{\alpha\gamma\theta}\quad n}} - {\overset{.}{\alpha}\quad A_{6}}} \right\}} + {m_{wn}B_{3}\left\{ {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{3n}S_{{\alpha\gamma\theta}\quad n}} - {\overset{.}{\alpha}A_{6}}} \right\}}} \right\rbrack} -} \\{\quad {{\overset{¨}{\alpha}m_{ba}A_{2}} + {{\overset{.}{\alpha}}^{2}m_{ba}A_{1}} - {{\overset{¨}{z}}_{6n}m_{sn}a_{1n}C_{{\alpha\gamma\eta}\quad n}} + {2{{\overset{.}{z}}_{6n}\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)}m_{sn}a_{1n}S_{{\alpha\gamma\eta}\quad n}} +}} \\{\quad {{{\overset{¨}{\eta}}_{n}m_{sn}z_{6n}a_{1n}S_{{\alpha\gamma\eta}\quad n}} + {{{\overset{.}{\eta}}_{n}\left( {{2\overset{.}{\alpha}} + {\overset{.}{\eta}}_{n}} \right)}m_{sn}z_{6n}a_{1n}C_{{\alpha\gamma\eta}\quad n}} - {{\overset{¨}{\theta}}_{n}m_{aw1n}a_{1n}C_{{\alpha\gamma\theta}\quad n}} +}} \\{\quad {{{{\overset{.}{\theta}}_{n}\left( {{2\overset{.}{\alpha}} + {\overset{.}{\theta}}_{n}} \right)}m_{aw1n}a_{1n}S_{{\alpha\gamma\theta}\quad n}} + {\overset{¨}{\alpha}a_{1n}\left\{ {{m_{sawcn}S_{{\alpha\gamma}\quad n}} - \quad {m_{sawbn}C_{\alpha}} +} \right.}}} \\{{\quad \left. {{m_{sn}z_{6n}S_{{\alpha\gamma\eta}\quad n}} - {m_{aw1n}C_{{\alpha\gamma\theta}\quad n}}} \right\}} - {{\overset{.}{\alpha}}^{2}a_{1n}\left\{ {{m_{sawcn}C_{{\alpha\gamma}\quad n}} + {m_{sawbn}S_{\alpha}} +} \right.}} \\{{\quad \left. {{m_{sn}z_{6n}C_{{\alpha\gamma\eta}\quad n}} + {m_{aw1n}S_{{\alpha\gamma\theta}\quad n}}} \right\}} - {{\overset{¨}{z}}_{0}\left\lbrack \left\{ {{m_{b}\left( {{b_{0}S_{\alpha}} + {c_{0}C_{\alpha}}} \right)} + {m_{awln}S_{{\alpha\gamma\eta}\quad n}} +} \right. \right.}} \\{\left. {{{\quad \left. {{z_{6n}m_{sn}C_{{\alpha\gamma\eta}\quad n}} + {m_{sawcn}C_{{\alpha\gamma}\quad n}} + {m_{sawbn}S_{\alpha}}} \right\}}S_{\beta}} + {\left( {m_{ba} + m_{sawan}} \right)C_{\beta}}} \right\rbrack +} \\{\quad {{{{\overset{.}{z}}_{0}\left( {1 - \overset{.}{\beta}} \right)}\left( {m_{ba} + m_{sawan}} \right)\sin \quad \beta} - {g\left\lbrack {{m_{ba}C_{\beta}} + {m_{b}A_{1}S_{\beta}} + \left\{ {{m_{sn}z_{6n}C_{{\alpha\gamma\eta}\quad n}} +} \right.} \right.}}} \\\left. {{{\quad \left. {{m_{aw1n}S_{{\alpha\gamma\theta}\quad n}} + {m_{sawcn}C_{{\alpha\gamma}\quad n}} + {m_{sawbn}S_{\alpha}}} \right\}}S_{\beta}} + {m_{sawan}C_{\beta}}} \right\rbrack \\{= \quad {\lambda_{3n}\left\lbrack {{{- \left\{ {{z_{12n}C_{\alpha}} + {e_{3n}S_{{\alpha\gamma\theta}\quad n}} + {c_{2n}C_{{\alpha\gamma}\quad n}} + {b_{2n}S_{\alpha}}} \right\}}S_{\beta}} + {a_{1n}C_{\beta}}} \right\rbrack}}\end{matrix} & (94) \\{\overset{¨}{\beta} = \frac{\begin{matrix}{2{\overset{.}{\beta}\left\lbrack {{\overset{.}{\alpha}m_{b}A_{1}A_{2}} + {m_{sn}B_{1}\left\{ {{{\overset{.}{z}}_{6n}C_{{\alpha\gamma\eta}\quad n}} - {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}S_{{\alpha\gamma\eta}\quad n}} - {\overset{.}{\alpha}A_{4}}} \right\}} +} \right.}} \\{\left. {{m_{an}B_{2}\left\{ {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{1n}C_{{\alpha\gamma\theta}\quad n}} - {\overset{.}{\alpha}A_{6}}} \right\}} + {m_{wn}B_{3}\left\{ {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{3n}S_{{\alpha\gamma\theta}\quad n}} - {\overset{.}{\alpha}A_{6}}} \right\}}} \right\rbrack -} \\{{\overset{¨}{\alpha}m_{ba}A_{2}} + {\overset{.}{\alpha}m_{ba}A_{1}} - {{\overset{¨}{z}}_{6n}m_{sn}a_{1n}C_{{\alpha\gamma\eta}\quad n}} + {2{{\overset{.}{z}}_{6n}\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)}m_{sn}\alpha_{1n}S_{{\alpha\gamma\eta}\quad n}} +} \\{{{\overset{¨}{\eta}}_{n}m_{sn}z_{6n}a_{1n}S_{{\alpha\gamma\eta}\quad n}} + {{{\overset{.}{\eta}}_{n}\left( {{2\overset{.}{\alpha}} + {\overset{.}{\eta}}_{n}} \right)}m_{sn}z_{6n}a_{1n}C_{{\alpha\gamma\eta}\quad n}} - {{\overset{¨}{\theta}}_{n}m_{aw1n}a_{1n}C_{{\alpha\gamma\theta}\quad n}} +} \\{{{{\overset{.}{\theta}}_{n}\left( {{2\overset{.}{\alpha}} + {\overset{.}{\theta}}_{n}} \right)}m_{aw1n}a_{1n}S_{{\alpha\gamma\theta}\quad n}} + {\overset{¨}{\alpha}a_{1n}\left\{ {{m_{sawcn}S_{{\alpha\gamma}\quad n}} - {m_{sawbn}C_{\alpha}} + {m_{sn}z_{6n}S_{{\alpha\gamma\eta}\quad n}} -} \right.}} \\{\left. {m_{aw1n}C_{{\alpha\gamma\theta}\quad n}} \right\} + {{\overset{.}{\alpha}}^{2}a_{1n}\left\{ {{m_{sawcn}C_{{\alpha\gamma}\quad n}} + {m_{sawbn}S_{\alpha}} + {m_{sn}z_{6n}C_{{\alpha\gamma\eta}\quad n}} + {m_{aw1n}S_{{\alpha\gamma\theta}\quad n}}} \right\}} -} \\{{\overset{¨}{z}}_{0}\left\lbrack \left\{ {{m_{b}\left( {{b_{0}S_{\alpha}} + {c_{0}C_{\alpha}}} \right)} + {m_{awln}S_{{\alpha\gamma\eta}\quad n}} + {z_{6n}m_{sn}C_{{\alpha\gamma\eta}\quad n}} + {m_{sawcn}C_{{\alpha\gamma}\quad n}} +} \right. \right.} \\{\left. {{\left. {m_{sawbn}S_{\alpha}} \right\} S_{\beta}} + {\left( {m_{ba} + m_{sawan}} \right)C_{\beta}}} \right\rbrack + {{{\overset{.}{z}}_{0}\left( {1 - \overset{.}{\beta}} \right)}\left( {m_{ba} + m_{sawan}} \right)\sin \quad \beta} -} \\{g\left\lbrack {{m_{ba}C_{\beta}} + {m_{b}A_{1}S_{\beta}} + \left\{ {{m_{sn}z_{6n}C_{{\alpha\gamma\eta}\quad n}} + {m_{aw1n}S_{{\alpha\gamma\theta}\quad n}} + {m_{sawcn}C_{{\alpha\gamma}\quad n}} +} \right.} \right.} \\{\left. {{\left. {m_{sawbn}S_{\alpha}} \right\} S_{\beta}} + {m_{sawan}C_{\beta}}} \right\rbrack + {\lambda_{3n}\left\{ \left( {{z_{12n}C_{\alpha}} + {e_{3n}S_{{\alpha\gamma\theta}\quad n}} + {c_{2n}C_{{\alpha\gamma}\quad n}} +} \right. \right.}} \\\left. {{\left. {b_{2n}S_{\alpha}} \right)S_{\beta}} - {a_{1n}C_{\beta}}} \right\}\end{matrix}}{- \left( {m_{saw2n} + m_{bal} + {m_{b}A_{1}^{2}} + {m_{sn}B_{1}^{2}} + {m_{an}B_{2}^{2}} + {m_{wn}B_{3}^{2}}} \right)}} & (95) \\{{{{\frac{\quad}{t}\left( \frac{\partial L}{\partial\overset{.}{\alpha}} \right)} - \frac{\partial L}{\partial\alpha}} = {{\frac{\partial F}{\partial\overset{.}{\alpha}} + {\sum\limits_{l,n}{\lambda_{1n}a_{1{n2}}\quad l}}} = 1}},2,{{3\quad n} = i},{ii},{iii},{iv}} & (96) \\\begin{matrix}{\quad {{{- \overset{¨}{\beta}}{m_{ba}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} + {\overset{.}{\beta}\overset{.}{\alpha}{m_{ba}\left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)}} +}} \\{\quad {{{\overset{¨}{z}}_{0}m_{b}\cos \quad {\beta \left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} - {\overset{.}{\beta}{\overset{.}{z}}_{0}m_{b}\sin \quad {\beta \left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} -}} \\{\quad {{\overset{.}{\alpha}{\overset{.}{z}}_{0}m_{b}\cos \quad {\beta \left( {{b_{0}\sin \quad \alpha} - {c_{0}\cos \quad \alpha}} \right)}} + {\overset{¨}{\alpha}{\langle{m_{b\quad b\quad l} + m_{saw1n} + {m_{sn}z_{6n}\left\lbrack {z_{6n} +} \right.}}}}}} \\{{{{\quad \left. {2\left\{ {{c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} \right\rbrack} - {2m_{aw1n}\left\{ {{c_{2n}\sin \quad \theta_{n}} - {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}}}\rangle} +} \\{\quad {\overset{.}{\alpha}{\langle{{m_{sn}{{\overset{.}{z}}_{6n}\left\lbrack {z_{6n} + {2\left\{ {{c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}}} \right\rbrack}} + {m_{sn}z_{6n}\left\lbrack {{\overset{.}{z}}_{6n} - {2{\overset{.}{\eta}}_{n}\left\{ {{c_{1n}\sin \quad \eta_{n}} +} \right.}} \right.}}}}} \\{{{\left. {\quad \left. {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}} \right\}} \right\rbrack - {2{\overset{.}{\theta}}_{n}m_{aw1n}\left\{ {{c_{2n}\cos \quad \theta_{n}} + {b_{2n}{\sin \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}}}\rangle} +} \\{\quad {{{\overset{¨}{z}}_{6n}m_{sn}\left\{ {{c_{1n}\sin \quad \eta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} + {{\overset{.}{z}}_{6n}{\overset{.}{\eta}}_{n}m_{sn}\left\{ {{c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}} \\{\quad {{{\overset{¨}{\eta}}_{n}m_{sn}z_{6n}\left\{ {z_{6n} + {c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} + {{\overset{.}{\eta}}_{n}m_{sn}{\overset{.}{z}}_{6n}\left\{ {z_{6n} + {c_{1n}\cos \quad \eta_{n}} -} \right.}}} \\{{\quad \left. {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}} \right\}} + {{\overset{.}{\eta}}_{n}m_{sn}z_{6n}\left\{ {{\overset{.}{z}}_{6n} - {{\overset{.}{\eta}}_{n}\left\lbrack {{c_{1n}\sin \quad \eta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\rbrack}} \right\}} +} \\{\quad {{\overset{¨}{\theta}\left\lbrack {m_{aw21n} - {m_{aw1n}\left\{ {{c_{2n}\sin \quad \theta_{n}} - {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}}} \right\rbrack} - {{\overset{.}{\theta}}_{n}^{2}m_{aw1n}\left\{ {{c_{2n}\cos \quad \theta_{n}} +} \right.}}} \\{\left. {\quad \left. {b_{2n}{\sin \left( {\gamma_{n} + \theta_{n}} \right)}} \right\}} \right\rbrack + {\overset{¨}{\beta}a_{1n}\left\{ {{m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} - {m_{sawbn}\cos \quad \alpha} +} \right.}} \\{{\quad \left. {{m_{sn}z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {m_{aw1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}}} \right\}} + {\overset{.}{\beta}a_{1n}\left\{ {\overset{.}{\alpha}\left\lbrack {{m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.} \right.}} \\{{\quad \left. {m_{sawbn}\sin \quad \alpha} \right\rbrack} + {m_{sn}{\overset{.}{z}}_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)m_{sn}z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \\{{\quad \left. {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{aw1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} \right\}} + {{\overset{¨}{z}}_{0}\left\{ {{m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -} \right.}} \\{\left. {{\quad \left. {{z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\cos \quad \alpha}} \right\}}\cos \quad \beta} \right\} +} \\{\quad {{\overset{.}{z}}_{0}\left\{ {{{- \left( {\alpha + {\overset{.}{\theta}}_{n}} \right)}m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -} \right.}} \\{{{\quad \left. {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {\overset{.}{\alpha}m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} - {\overset{.}{\alpha}m_{sawbn}\sin \quad \alpha}} \right\}}\cos \quad \beta} -} \\{\quad {\overset{.}{\beta}{\overset{.}{z}}_{0}\left\{ {{m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{\left. {{\quad \left. {m_{sawbn}\cos \quad \alpha} \right\}}\cos \quad \beta} \right\} - {\left\{ {{{\overset{.}{\beta}}^{2}{m_{b}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}} + {\overset{.}{\alpha}\overset{.}{\beta}m_{ba}}} \right\} \left( {{b_{0}\sin \quad \alpha} + {c_{0}\cos \quad \alpha}} \right)} -} \\{\quad \left| {\overset{.}{\beta}{\langle{m_{sn}\left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} \left\{ {{{- z_{6n}}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -} \right.}}} \right.} \\{{\quad \left. {{c_{1n}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\cos \quad \alpha}} \right\}} + {m_{an}\left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}\left\{ {{e_{1}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\cos \quad \alpha}} \right\}} +} \\{\quad {m_{wn}\left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} \left\{ {{e_{3}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -} \right.}} \\{{{\quad \left. {{c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\cos \quad \alpha}} \right\}}\rangle} + {{\overset{.}{z}}_{6n}\overset{.}{\beta}m_{sn}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \\{\quad {{{\overset{.}{\eta}}_{n}\overset{.}{\beta}m_{sn}z_{6n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {\overset{.}{\theta}\overset{.}{\beta}m_{aw1n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +}} \\{\quad {\overset{.}{\alpha}\overset{.}{\beta}a_{1n}\left\{ {{m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {m_{sawbn}\sin \quad \alpha} + {m_{sn}z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right.}} \\{{\quad \left. {m_{aw1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} \right\}} - {{\overset{.}{z}}_{0}\left\{ {{{\overset{.}{z}}_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +} \right.}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{awln}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +}} \\{{\quad \left. {{\overset{.}{\alpha}m_{sawcn}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {\overset{.}{\alpha}m_{sawbn}\sin \quad \alpha}} \right\}}\cos \quad \beta} \\{\quad {\beta {\overset{.}{z}}_{0}\left\lbrack \left\{ {{m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {m_{sawn}{\sin \left( {\alpha + \gamma_{n}} \right)}} +} \right. \right.}} \\\left. {{\quad \left. {m_{sawbn}\cos \quad \alpha} \right\}}\sin \quad \beta} \middle| {{{+ {{gm}_{b}\left( {{b_{0}\cos \quad \alpha} - {c_{0}\sin \quad \alpha}} \right)}}\cos \quad \beta} -} \right. \\{\quad {g\left\{ {{m_{sn}z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {m_{aw1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {m_{sawcn}{\sin \left( {\alpha + \gamma_{n}} \right)}} -} \right.}} \\{{\quad \left. {m_{sawbn}\cos \quad \alpha} \right\}}\cos \quad \beta} \\{= \quad {\lambda_{3n}\left\{ {{{- z_{12n}}\sin \quad \alpha} + {e_{3n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\cos \quad \alpha}} \right\} \cos \quad \beta}}\end{matrix} & (97) \\\begin{matrix}{\quad {{{\overset{¨}{z}}_{0}\left\{ {{m_{b}A_{2}} + {m_{awln}C_{{\alpha\gamma\theta}\quad n}} - {z_{6n}m_{sn}S_{{\alpha\gamma\eta}\quad n}} - {m_{sawcn}S_{{\alpha\gamma}\quad n}} + {m_{sawbn}C_{\alpha}}} \right\} C_{\beta}} -}} \\{\quad {{\overset{¨}{\beta}m_{ba}A_{2}} + {\overset{¨}{\alpha}\left\{ {m_{b\quad b\quad l} + m_{saw1n} + {m_{sn}{z_{6n}\left( {z_{6n} + {2E_{1n}}} \right)}} - {2m_{aw1n}H_{1n}}} \right\}} +}} \\{\quad {{2\overset{.}{\alpha}\left\{ {{m_{sn}{{\overset{.}{z}}_{6n}\left( {z_{6n} + E_{1n}} \right)}} - {m_{sn}z_{6n}{\overset{.}{\eta}}_{n}E_{2n}} - {{\overset{.}{\theta}}_{n}m_{aw1n}H_{2n}}} \right\}} + {{\overset{¨}{z}}_{6n}m_{sn}E_{2n}} +}} \\{\quad {{{\overset{.}{z}}_{6n}{\overset{.}{\eta}}_{n}m_{sn}E_{1n}} + {{\overset{¨}{\eta}}_{n}m_{sn}z_{6n}\left\{ {z_{6n} + E_{1n}} \right\}} + {{\overset{.}{\eta}}_{n}m_{sn}{\overset{.}{z}}_{6n}\left\{ {{2z_{6n}} + E_{1n}} \right\}} -}} \\{\quad {{{\overset{.}{\eta}}_{n}^{2}m_{sn}z_{6n}E_{2n}} + {\overset{¨}{\theta}\left( {m_{aw21n}H_{1n}} \right)} - {{\overset{.}{\theta}}_{n}^{2}m_{aw1n}H_{2n}} + {\overset{¨}{\beta}a_{1n}\left( {{m_{sawcn}S_{{\alpha\gamma}\quad n}} - {m_{sawbn}C_{\alpha}} +} \right.}}} \\{{\quad \left. {{m_{sn}z_{6n}S_{{\alpha\gamma\eta}\quad n}} - {m_{aw1n}C_{{\alpha\gamma\theta}\quad n}}} \right)} + {\overset{.}{\beta}a_{1n}\left\{ {{\overset{.}{\alpha}\left( {{m_{sawcn}C_{{\alpha\gamma}\quad n}} + {m_{sawbn}S_{\alpha}}} \right)} +} \right.}} \\{{\quad \left. {{m_{sn}{\overset{.}{z}}_{6n}S_{{\alpha\gamma\eta}\quad n}} + {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)m_{sn}z_{6n}C_{{\alpha\gamma\eta}\quad n}} + {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)m_{aw1n}S_{{\alpha\gamma\theta}\quad n}}} \right\}} -} \\{\quad {{{\overset{.}{\beta}}^{2}m_{b}A_{2}A_{1}} - \left\lbrack {{\overset{.}{\beta}}^{2}\left\{ {{m_{sn}{B_{1}\left( {{{- z_{6n}}S_{{\alpha\gamma\eta}\quad n}} - A_{4}} \right)}} + {m_{an}{B_{2}\left( {{e_{1}C_{{\alpha\gamma\theta}\quad n}} - A_{6}} \right)}} +} \right.} \right.}} \\{{\quad \left. {m_{wn}{B_{3}\left( {{e_{3}C_{{\alpha\gamma\theta}\quad n}} - A_{6}} \right)}} \right\}} + {{\overset{.}{z}}_{6n}\overset{.}{\beta}m_{sn}a_{1n}S_{{\alpha\gamma\eta}\quad n}} + {{\overset{.}{\eta}}_{n}\overset{.}{\beta}m_{sn}z_{6n}a_{1n}C_{{\alpha\gamma\eta}\quad n}} +} \\{\quad {{\overset{.}{\theta}\overset{.}{\beta}m_{{\alpha\gamma\eta}\quad n}a_{1n}S_{{\alpha\gamma\theta}\quad n}} + {\overset{.}{\alpha}\overset{.}{\beta}a_{1n}\left\{ {{m_{sawcn}C_{{\alpha\gamma}\quad n}} + {m_{sawbn}S_{a}} +} \right.}}} \\{\left. {\quad \left. {{m_{sn}z_{6n}C_{{\alpha\gamma\eta}\quad n}} + {m_{aw1n}S_{{\alpha\gamma\theta}\quad n}}} \right\}} \right\rbrack + {{gm}_{b}A_{2}C_{\beta}} - {g\left\{ {{m_{sn}z_{6n}S_{{\alpha\gamma\eta}\quad n}} -} \right.}} \\{{\quad \left. {{m_{aw1n}C_{{\alpha\gamma\theta}\quad n}} + {m_{sawcn}S_{{\alpha\gamma}\quad n}} - {m_{sawbn}C_{\alpha}}} \right\}}C_{\beta}} \\{= \quad {\lambda_{3n}\left\{ {{{- z_{12n}}S_{\alpha}} + {e_{3n}C_{{\alpha\gamma\theta}\quad n}} - {c_{2n}S_{{\alpha\gamma}\quad n}} + {b_{2n}C_{a}}} \right\} C_{\beta}}}\end{matrix} & (98) \\\begin{matrix}{\quad {{{\overset{¨}{z}}_{0}\left\{ {{m_{b}A_{2}} + {m_{awln}C_{{\alpha\gamma\theta}\quad n}} - {z_{6n}m_{sn}S_{{\alpha\gamma\eta}\quad n}} - {m_{sawcn}S_{{\alpha\gamma}\quad n}} + {m_{sawbn}C_{\alpha}}} \right\} C_{\beta}} -}} \\{\quad {{\overset{¨}{\beta}m_{ba}A_{2}} + {\overset{¨}{\alpha}\left\{ {m_{b\quad b\quad l} + m_{saw1n} + {m_{sn}{z_{6n}\left( {z_{6n} + {2E_{1n}}} \right)}} - {2m_{aw1n}H_{1n}}} \right\}} +}} \\{\quad {{{m_{sn}\left( {{2\overset{.}{\alpha}{\overset{.}{z}}_{6n}} + {{\overset{¨}{\eta}}_{n}z_{6n}} + {2{\overset{.}{\eta}}_{n}{\overset{.}{z}}_{6n}}} \right)}\left( {z_{6n} + E_{1n}} \right)} - {2{\overset{.}{\alpha}\left( {{m_{sn}z_{6n}{\overset{.}{\eta}}_{n}E_{2n}} + {{\overset{.}{\theta}}_{n}m_{aw1n}H_{2n}}} \right)}} +}} \\{\quad {{{\overset{¨}{z}}_{6n}m_{sn}E_{2n}} - {{\overset{.}{\eta}}_{n}^{2}m_{sn}z_{6n}E_{2n}} + {\overset{¨}{\theta}\left( {m_{aw2ln} - {m_{aw1n}H_{1n}}} \right)} - {{\overset{.}{\theta}}_{n}^{2}m_{aw1n}H_{2n}} +}} \\{\quad {{\overset{¨}{\beta}{a_{1n}\left( {{m_{sawcn}S_{{\alpha\gamma}\quad n}} - {m_{sawbn}C_{\alpha}} + {m_{sn}z_{6n}S_{{\alpha\gamma\eta}\quad n}} - {m_{aw1n}C_{{\alpha\gamma\theta}\quad n}}} \right)}} -}} \\{\quad {{\overset{.}{\beta}}^{2}\left\{ {{m_{b}A_{2}A_{1}} + {m_{sn}{B_{1}\left( {{{- z_{6n}}S_{{\alpha\gamma\eta}\quad n}} - A_{4}} \right)}} + {m_{an}{B_{2}\left( {{e_{1}C_{{\alpha\gamma\theta}\quad n}} - A_{6}} \right)}} +} \right.}} \\{{\quad \left. {m_{wn}{B_{3}\left( {{e_{3}C_{{\alpha\gamma\theta}\quad n}} - A_{6}} \right)}} \right\}} + {{gm}_{b}A_{2}C_{\beta}} - {g\left\{ {{m_{sn}z_{6n}S_{{\alpha\gamma\eta}\quad n}} - {m_{aw1n}C_{{\alpha\gamma\theta}\quad n}} +} \right.}} \\{{\quad \left. {{m_{sawcn}S_{{\alpha\gamma}\quad n}} - {m_{sawbn}C_{\alpha}}} \right\}}C_{\beta}} \\{= \quad {{\lambda_{3n}\left( {{{- z_{12n}}S_{\alpha}} + {e_{3n}C_{{\alpha\gamma\theta}\quad n}} - {c_{2n}S_{{\alpha\gamma}\quad n}} + {b_{2n}C_{\alpha}}} \right)}C_{\beta}}}\end{matrix} & (99) \\\frac{{\therefore\overset{¨}{a}} = \begin{matrix}{{\overset{¨}{z}}_{0}\left\{ {{m_{b}A_{2}} + {m_{awln}C_{{\alpha\gamma\theta}\quad n}} - {z_{6n}m_{sn}S_{{\alpha\gamma\eta}\quad n}} - {m_{sawcn}S_{{\alpha\gamma}\quad n}} + {m_{sawbn}C_{\alpha}}} \right\} C_{\beta}} \\{{{m_{sn}\left( {{2\overset{.}{\alpha}{\overset{.}{z}}_{6n}} + {{\overset{¨}{\eta}}_{n}z_{6n}} + {2{\overset{.}{\eta}}_{n}{\overset{.}{z}}_{6n}}} \right)}\left( {z_{6n} + E_{1n}} \right)} - {2{\overset{.}{\alpha}\left( {{m_{sn}z_{6n}{\overset{.}{\eta}}_{n}E_{2n}} + {{\overset{.}{\theta}}_{n}m_{aw1n}H_{2n}}} \right)}} +} \\{{{\overset{¨}{z}}_{6n}m_{sn}E_{2n}} - {{\overset{.}{\eta}}_{n}^{2}m_{sn}z_{6n}E_{2n}} + {\overset{¨}{\theta}\left( {m_{aw21n} - {m_{aw1n}H_{1n}}} \right)} - {{\overset{.}{\theta}}_{n}^{2}m_{aw1n}H_{2n}} +} \\{{\overset{¨}{\beta}{a_{1n}\left( {{m_{sawcn}S_{{\alpha\gamma}\quad n}} - {m_{sawbn}C_{\alpha}} + {m_{sn}z_{6n}S_{{\alpha\gamma\eta}\quad n}} - {m_{aw1n}C_{{\alpha\gamma\theta}\quad l}}} \right)}} - {{\overset{.}{\beta}}^{2}\left\{ {{m_{b}A_{2}A_{1}} +} \right.}} \\{{m_{sn}{B_{1}\left( {{{- z_{6n}}S_{{\alpha\gamma\eta}\quad n}} - A_{4}} \right)}} + {m_{an}{B_{2}\left( {{e_{1}C_{{\alpha\gamma\theta}\quad n}} - A_{6}} \right)}} + {m_{wn}B_{3}\left( {{e_{3}C_{{\alpha\gamma\theta}\quad n}} -} \right.}} \\{\left. \left. A_{6} \right) \right\} + {{gm}_{b}A_{2}C_{\beta}} - {g\left\{ {{m_{sn}z_{6n}S_{{\alpha\gamma\eta}\quad n}} - {m_{aw1n}C_{{\alpha\gamma\theta}\quad n}} + {m_{sawcn}S_{{\alpha\gamma}\quad n}} -} \right.}} \\{{\left. {m_{sawbn}C_{\alpha}} \right\} C_{\beta}} - {\overset{¨}{\beta}m_{ba}A_{2}} + {{\lambda_{3n}\left( {{z_{12n}S_{\alpha}} - {e_{3n}C_{{\alpha\gamma\theta}\quad n}} + {c_{2n}S_{{\alpha\gamma}\quad n}} - {b_{2n}C_{\alpha}}} \right)}C_{\beta}}}\end{matrix}}{- \left\{ {m_{b\quad b\quad l} + m_{saw1n} + {m_{sn}{z_{6n}\left( {z_{6n} + {2E_{1n}}} \right)}} - {2m_{aw1n}H_{1n}}} \right\}} & (100) \\{{{{\frac{\quad}{t}\left( \frac{\partial L}{\partial{\overset{.}{\eta}}_{n}} \right)} - \frac{\partial L}{\partial\eta_{n}}} = {{\frac{\partial F}{\partial{\overset{.}{\eta}}_{n}} + {\sum\limits_{l,n}{\lambda_{l\quad n}a_{i\quad {n3}}\quad l}}} = 1}},2,{{3\quad n} = i},{ii},{iii},{iv}} & (101) \\\begin{matrix}{\quad {{m_{sn}{\overset{¨}{\eta}}_{n}z_{6n}^{2}} + {2m_{sn}{\overset{.}{\eta}}_{n}{\overset{.}{z}}_{6n}z_{6n}} + {\overset{¨}{\alpha}m_{sn}z_{6n}\left\{ {z_{6n} + {c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}\quad} \\{\quad {{\overset{.}{\alpha}m_{sn}{\overset{.}{z}}_{6n}\left\{ {z_{6n} + {c_{1n}\cos \quad \eta_{n}} - {b_{2}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} + {\overset{.}{\alpha}m_{sn}z_{6n}\left\{ {{\overset{.}{z}}_{6n} - {{\overset{.}{\eta}}_{n}\left\lbrack {{c_{1n}\sin \quad \eta_{n}} +} \right.}} \right.}}} \\{\left. {\quad \left. {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}} \right\rbrack} \right\} + {\overset{¨}{\beta}m_{sn}z_{6n}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {\overset{.}{\beta}m_{sn}{\overset{.}{z}}_{6n}a_{1n}{\sin\left( {\alpha + \gamma_{n} +} \right.}}} \\{{\quad \left. \eta_{n} \right)} + {{\overset{.}{\beta}\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)}m_{sn}z_{6n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} - {{\overset{¨}{z}}_{0}z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} -} \\{\quad {{{\overset{.}{z}}_{0}{\overset{.}{z}}_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} - {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right){\overset{.}{z}}_{0}z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} -}} \\{\quad {{\overset{.}{\beta}{\overset{.}{z}}_{0}{\overset{.}{z}}_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\sin \quad \beta} - {\langle{{{\overset{.}{\alpha}}^{2}m_{sn}z_{6n}\left\{ {{{- c_{1n}}\sin \quad \eta_{n}} - {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +}}}} \\{\quad {{\overset{.}{\beta}}^{2}m_{sn}\left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} \left\{ {{- z_{6n}}{\sin\left( {\alpha + \gamma_{n} +} \right.}} \right.}} \\{\left. {\quad \left. \eta_{n} \right)} \right\} + {{\overset{.}{z}}_{6n}\overset{.}{\alpha}m_{sn}\left\{ {{c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} + {{\overset{.}{z}}_{6n}\overset{.}{\beta}m_{sn}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -} \\{\quad {{{\overset{.}{\eta}}_{n}\overset{.}{\alpha}m_{sn}z_{6n}\left\{ {{c_{1n}\sin \quad \eta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} + {{\overset{.}{\eta}}_{n}\overset{.}{\beta}m_{sn}z_{6n}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} +}} \\{\quad {{\overset{.}{\alpha}\overset{.}{\beta}a_{1n}m_{sn}z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {{gm}_{sn}z_{6n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} -}} \\{\quad {{{\overset{.}{z}}_{0}\left\{ {{z_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right){\overset{.}{z}}_{0}z_{6n}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}}} \right\} \cos \quad \beta} +}} \\{\quad {{\overset{.}{\beta}{\overset{.}{z}}_{0}{\overset{.}{z}}_{6n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\sin \quad \beta}\rangle}} \\{= \quad {{{\lambda_{1n}\left( {z_{6n} - d_{1n}} \right)}\cos \quad \eta_{n}} + {{\lambda_{2n}\left( {z_{6n} - d_{1n}} \right)}\sin \quad \eta_{n}}}}\end{matrix} & (102) \\\begin{matrix}{\quad {{m_{sn}{\overset{¨}{\eta}}_{n}z_{6n}^{2}} + {2m_{sn}{\overset{.}{\eta}}_{n}{\overset{.}{z}}_{6n}z_{6n}} + {\overset{¨}{\alpha}m_{sn}z_{6n}\left\{ {z_{6n} + E_{1}} \right\}} + {\overset{.}{\alpha}m_{sn}{\overset{.}{z}}_{6n}\left\{ {{2z_{6n}} + E_{1}} \right\}} -}} \\{\quad {{\overset{.}{\alpha}m_{sn}z_{6n}{\overset{.}{\eta}}_{n}E_{2}} + {\overset{¨}{\beta}m_{sn}z_{6n}a_{1n}S_{{\alpha\gamma\eta}\quad n}} + {\overset{.}{\beta}m_{sn}{\overset{.}{z}}_{6n}a_{1n}S_{{\alpha\gamma\eta}\quad n}} +}} \\{\quad {{{\overset{.}{\beta}\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)}m_{sn}z_{6n}a_{1_{n}}C_{{\alpha\gamma\eta}\quad n}} - {{\overset{¨}{z}}_{0}z_{6n}m_{sn}S_{{\alpha\gamma\eta}\quad n}C_{\beta}} + {{\overset{.}{\alpha}}^{2}m_{sn}z_{6n}E_{2}} +}} \\{\quad {{{\overset{.}{\beta}}^{2}m_{sn}B_{1}z_{6n}S_{{\alpha\gamma\eta}\quad n}} - {{\overset{.}{z}}_{6n}\overset{.}{\alpha}m_{sn}E_{1}} - {{\overset{.}{z}}_{6n}\overset{.}{\beta}m_{sn}a_{1n}S_{{\alpha\gamma\eta}\quad n}} + {{\overset{.}{\eta}}_{n}\overset{.}{\alpha}m_{sn}z_{6n}E_{2}} -}} \\{\quad {{{\overset{.}{\eta}}_{n}\overset{.}{\beta}m_{sn}z_{6n}a_{1n}C_{{\alpha\gamma\eta}\quad n}} - {\overset{.}{\alpha}\overset{.}{\beta}a_{1n}m_{sn}z_{6n}C_{{\alpha\gamma\eta}\quad n}} - {{gm}_{sn}z_{6n}S_{{\alpha\gamma\eta}\quad n}C_{\beta}}}} \\{= \quad {{{- {\lambda_{1n}\left( {z_{6n} - d_{1n}} \right)}}C_{\eta \quad n}} + {{\lambda_{2n}\left( {z_{6n} - d_{1n}} \right)}S_{\eta \quad n}}}}\end{matrix} & (103) \\\begin{matrix}{\quad {m_{sn}z_{6n}\left\{ {{{\overset{¨}{\eta}}_{n}z_{6n}} + {2{\overset{.}{\eta}}_{n}{\overset{.}{z}}_{6n}} + {\overset{¨}{\alpha}\left( {z_{6n} + E_{1}} \right)} + {2\overset{.}{\alpha}{\overset{.}{z}}_{6n}} + {\overset{¨}{\beta}a_{1n}S_{{\alpha\gamma\eta}\quad n}} -} \right.}} \\{\quad \left. {{{\overset{¨}{z}}_{0}S_{{\alpha\gamma\eta}\quad n}C_{\beta}} + {{\overset{.}{\alpha}}^{2}E_{2}} + {{\overset{.}{\beta}}^{2}B_{1}S_{{\alpha\gamma\eta}\quad n}} - {{gS}_{{\alpha\gamma\eta}\quad n}C_{\beta}}} \right\}} \\{= \quad {{{- {\lambda_{1n}\left( {z_{6n} - d_{1n}} \right)}}C_{\eta \quad n}} + {{\lambda_{2n}\left( {z_{6n} - d_{1n}} \right)}S_{\eta \quad n}}}}\end{matrix} & (104) \\{{\therefore\lambda_{1n}} = \frac{\begin{matrix}{m_{sn}z_{6n}\left\{ {{{\overset{¨}{\eta}}_{n}z_{6n}} + {2{\overset{.}{\eta}}_{n}{\overset{.}{z}}_{6n}} + {\overset{¨}{\alpha}\left( {z_{6n} + E_{1}} \right)} + {2\overset{.}{\alpha}{\overset{.}{z}}_{6n}} + {\overset{¨}{\beta}a_{1n}S_{{\alpha\gamma\eta}\quad n}} -} \right.} \\{\left. {{{\overset{¨}{z}}_{0}S_{{\alpha\gamma\eta}\quad n}C_{\beta}} + {{\overset{.}{\alpha}}^{2}E_{2}} + {{\overset{.}{\beta}}^{2}B_{1}S_{{\alpha\gamma\eta}\quad n}} - {{gS}_{{\alpha\gamma\eta}\quad n}C_{\beta}}} \right\} - {{\lambda_{2n}\left( {z_{6n} - d_{1n}} \right)}S_{\eta \quad n}}}\end{matrix}}{{- \left( {z_{6n} - d_{1n}} \right)}C_{\eta \quad n}}} & (105) \\{{{{\frac{\quad}{t}\left( \frac{\partial L}{\partial{\overset{.}{\theta}}_{n}} \right)} - \frac{\partial L}{\partial\theta_{n}}} = {{\frac{\partial F}{\partial{\overset{.}{\theta}}_{n}} + {\sum\limits_{l,n}{\lambda_{1n}a_{1{n4}}\quad l}}} = 1}},2,{{3\quad n} = i},{ii},{iii},{iv}} & (106) \\\begin{matrix}{\quad {{{\overset{¨}{\theta}}_{n}m_{aw2ln}} + {\overset{¨}{\alpha}\left\lbrack {m_{aw2ln} - {m_{aw1n}\left\{ {{c_{2}\sin \quad \theta_{n}} - {b_{2n}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}} -} \right.}}} \\{\quad {{\overset{.}{\alpha}{\overset{.}{\theta}}_{n}m_{aw1n}\left\{ {{c_{2n}\cos \quad \theta_{n}} + {b_{2n}{\sin \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}} - {\overset{¨}{\beta}m_{aw1n}{a_{1n}\left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +}} \\{\quad {{{\overset{.}{\beta}\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)}m_{aw1n}a_{1_{n}}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {{\overset{¨}{z}}_{0}m_{awln}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\cos \quad \beta} -}} \\{\quad {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right){\overset{.}{z}}_{0}m_{aw1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\cos \quad \beta} - {\overset{.}{\beta}{\overset{.}{z}}_{0}m_{aw1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\sin \quad \beta} -}} \\{\quad \left\lbrack {{{- k_{zi}}e_{0i}^{2}\left\{ {{\sin \left( {\gamma_{i} + \theta_{i}} \right)} - {\sin \left( {\gamma_{ii} + \theta_{ii}} \right)}} \right\} {\cos \left( {\gamma_{n} + \theta_{n}} \right)}X_{s}} - {k_{ziii}e_{0{iii}}^{2}\left\{ {\sin\left( {\gamma_{iii} +} \right.} \right.}} \right.} \\{{\left. {{\quad \left. \theta_{iii} \right)} - {\sin \left( {\gamma_{iv} + \theta_{iv}} \right)}} \right\} {\cos \left( {\gamma_{n} + \theta_{n}} \right)}X_{s}} - {{\overset{.}{\alpha}}^{2}m_{aw1n}\left\{ {{c_{2n}\cos \quad \theta_{n}} + {b_{2n}{\sin \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}} +} \\{\quad {{\overset{.}{\beta}}^{2}{\langle{{m_{an}\left\{ {{e_{1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} e_{1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +}}}} \\{{\quad {{m_{wn}\left\{ {{e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} e_{3n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}}\rangle}} -} \\{\quad {{\overset{.}{\theta}\overset{.}{\alpha}m_{aw1n}\left\{ {{c_{2n}\cos \quad \theta_{n}} + {b_{2n}{\sin \left( {\gamma_{n} + \theta_{n}} \right)}}} \right\}} + {\overset{.}{\theta}\overset{.}{\beta}m_{aw1n}a_{1_{n}}\sin \quad \left( {\alpha + \gamma_{n} + \theta_{n}} \right)} +}} \\{\quad {{\overset{.}{\alpha}\overset{.}{\beta}a_{1n}m_{aw1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {{gm}_{aw1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\cos \quad \beta} -}} \\{\quad \left. {{{{\overset{.}{z}}_{0}\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)}m_{aw1n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\cos \quad \beta} - {\overset{.}{\beta}{\overset{.}{z}}_{0}m_{aw1n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\sin \quad \beta}} \right\rbrack} \\{= \quad {{\lambda_{1n}e_{2n}\sin \quad \theta_{n}} + {\lambda_{2n}e_{2n}\cos \quad \theta_{n}} + {\lambda_{3n}e_{3n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\cos \quad \beta}}}\end{matrix} & (107) \\\begin{matrix}{\quad {{{\overset{¨}{\theta}}_{n}m_{aw2ln}} + {\overset{¨}{\alpha}\left( {m_{aw2ln} - {m_{aw1n}H_{1}}} \right)} - {\overset{.}{\alpha}{\overset{.}{\theta}}_{n}m_{aw1n}H_{2}} - {\overset{¨}{\beta}m_{aw1n}a_{1n}C_{{\alpha\gamma\theta}\quad n}} +}} \\{\quad {{{\overset{.}{\beta}\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)}m_{aw1n}a_{1n}S_{{\alpha\gamma\theta}\quad n}} + {{\overset{¨}{z}}_{0}m_{awln}C_{{\alpha\gamma\theta}\quad n}C_{\beta}} - \left\lbrack {{- k_{zi}}e_{0i}^{2}\left\{ {{\sin \left( {\gamma_{i} + \theta_{i}} \right)} +} \right.} \right.}} \\{{{\quad \left. {\sin \left( {\gamma_{ii} + \theta_{ii}} \right)} \right\}}X_{s}} - {k_{ziii}e_{0{iii}}^{2}\left\{ {{\sin \left( {\gamma_{iii} + \theta_{iii}} \right)} + {\sin \left( {\gamma_{iv} + \theta_{iv}} \right)}} \right\} {\cos \left( {\gamma_{n} + \theta_{n}} \right)}X_{s}} -} \\{\quad {{{\overset{.}{\alpha}}^{2}m_{aw1n}H_{2}} + {{\overset{.}{\beta}}^{2}\left( {{m_{an}B_{2}e_{1n}C_{{\alpha\gamma\theta}\quad n}} + {m_{wn}B_{3}e_{3n}C_{{\alpha\gamma\theta}\quad n}}} \right)} - {\overset{.}{\theta}\overset{.}{\alpha}m_{aw1n}H_{2}} +}} \\{\quad \left. {{\overset{.}{\theta}\overset{.}{\beta}m_{aw1n}a_{1n}S_{{\alpha\gamma\theta}\quad n}} + {\overset{.}{\alpha}\overset{.}{\beta}a_{1n}m_{aw1n}S_{{\alpha\gamma\theta}\quad n}} - {{gm}_{aw1n}C_{{\alpha\gamma\theta}\quad n}C_{\beta}}} \right\rbrack} \\{= \quad {{\lambda_{1n}e_{2n}S_{\theta \quad n}} + {\lambda_{2n}e_{2n}C_{\theta \quad n}} + {\lambda_{3n}e_{3n}C_{{\alpha\gamma\theta}\quad n}C_{\beta}}}}\end{matrix} & (108) \\\begin{matrix}{\quad {{{\overset{¨}{\theta}}_{n}m_{aw2ln}} + {\overset{¨}{\alpha}\left( {m_{aw2ln} - {m_{aw1n}H_{1}}} \right)} - {\overset{¨}{\beta}m_{aw1n}a_{1n}C_{{\alpha\gamma\theta}\quad n}} + {{\overset{¨}{z}}_{0}m_{awln}C_{{\alpha\gamma\theta}\quad n}C_{\beta}} +}} \\{\quad {{{\overset{.}{\alpha}}^{2}m_{aw1n}H_{2}} - {{\overset{.}{\beta}}^{2}\left( {{m_{an}B_{2}e_{1n}C_{{\alpha\gamma\theta}\quad n}} + {m_{wn}B_{3}e_{3n}C_{{\alpha\gamma\theta}\quad n}}} \right)} + {{gm}_{aw1n}C_{{\alpha\gamma\theta}\quad n}C_{\beta}} +}} \\{\quad {{k_{zi}e_{0i}^{2}\left\{ {{\sin \left( {\gamma_{i} + \theta_{i}} \right)} + {\sin \left( {\gamma_{ii} + \theta_{ii}} \right)}} \right\} {\cos \left( {\gamma_{n} + \theta_{n}} \right)}} + {k_{ziii}e_{0{iii}}^{2}\left\{ {{\sin \left( {\gamma_{iii} + \theta_{iii}} \right)} +} \right.}}} \\{{\quad \left. {\sin \left( {\gamma_{iv} + \theta_{iv}} \right)} \right\}}{\cos \left( {\gamma_{n} + \theta_{n}} \right)}} \\{= \quad {{\lambda_{1n}e_{2n}S_{\theta \quad n}} + {\lambda_{2n}e_{2n}C_{\theta \quad n}} + {\lambda_{3n}e_{3n}C_{{\alpha\gamma\theta}\quad n}C_{\beta}}}}\end{matrix} & (109) \\{{\overset{¨}{\theta}}_{n} = \frac{\begin{matrix}{\therefore{{\overset{¨}{\alpha}\left( {m_{aw2ln}H_{1}} \right)} - {\overset{¨}{\beta}m_{aw1n}a_{1n}C_{{\alpha\gamma\theta}\quad n}} + {{\overset{¨}{z}}_{0}m_{awln}C_{{\alpha\gamma\theta}\quad n}C_{\beta}} +}} \\{{{\overset{.}{\alpha}}^{2}m_{aw1n}H_{2}} - {{\overset{.}{\beta}}^{2}\left( {{m_{an}B_{2}e_{1n}C_{{\alpha\gamma\theta}\quad n}} + {m_{wn}B_{3}e_{3n}C_{{\alpha\gamma\theta}\quad n}}} \right)} + {{gm}_{aw1n}C_{{\alpha\gamma\theta}\quad n}C_{\beta}} -} \\{{\lambda_{1n}e_{2n}S_{\theta \quad n}} - {\lambda_{2n}e_{2n}C_{\theta \quad n}} - {\lambda_{3n}e_{3n}C_{{\alpha\gamma\theta}\quad n}C_{\beta}} + {k_{zi}e_{0i}^{2}\left\{ {{\sin \left( {\gamma_{i} + \theta_{i}} \right)} +} \right.}} \\{{\left. {\sin \left( {\gamma_{ii} + \theta_{ii}} \right)} \right\} {\cos \left( {\gamma_{n} + \theta_{n}} \right)}} + {k_{ziii}e_{0{iii}}^{2}\left\{ {{\sin \left( {\gamma_{iii} + \theta_{iii}} \right)} +} \right.}} \\{\left. {\sin \left( {\gamma_{iv} + \theta_{iv}} \right)} \right\} {\cos \left( {\gamma_{n} + \theta_{n}} \right)}}\end{matrix}}{- m_{aw21n}}} & (110) \\{{{{\frac{\quad}{t}\left( \frac{\partial L}{\partial{\overset{.}{z}}_{6n}} \right)} - \frac{\partial L}{\partial z_{6n}}} = {{\frac{\partial F}{\partial{\overset{.}{z}}_{6n}} + {\sum\limits_{l,n}{\lambda_{1n}a_{1{n5}}\quad l}}} = 1}},2,{{3\quad n} = i},{ii},{iii},{iv}} & (111) \\\begin{matrix}{\quad {{m_{sn}{\overset{¨}{z}}_{6n}} + {\overset{¨}{\alpha}m_{sn}\left\{ {{c_{1n}\sin \quad \eta_{n}} + {b_{2n}{\cos \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} + {\overset{.}{\alpha}{\overset{.}{\eta}}_{n}m_{sn}\left\{ {{c_{1n}\cos \quad \eta_{n}} -} \right.}}} \\{{\quad \left. {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}} \right\}} - {\overset{¨}{\beta}m_{sn}a_{1n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {{\overset{.}{\beta}\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)}m_{sn}a_{1n}{\sin\left( {\alpha +} \right.}}} \\{{\quad \left. {\gamma_{n} + \eta_{n}} \right)} + {{\overset{¨}{z}}_{0}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} - {\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right){\overset{.}{z}}_{0}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} -} \\{\quad {{\overset{.}{\beta}{\overset{.}{z}}_{0}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\sin \quad \beta} - {\langle{{m_{sn}{\overset{.}{\eta}}_{n}^{2}z_{6n}} + {{\overset{.}{\alpha}}^{2}{m_{sn}\left\lbrack {z_{6n} + \left\{ {{c_{1n}\cos \quad \eta_{n}} -} \right.} \right.}}}}}} \\{\left. {\quad \left. {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}} \right\}} \right\rbrack + {{\overset{.}{\beta}}^{2}m_{sn}\left\{ {{z_{6n}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {c_{1n}{\cos \left( {\alpha + \gamma_{n}} \right)}} +} \right.}} \\{{{\quad \left. {b_{2n}\sin \quad \alpha} \right\}}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {{\overset{.}{\eta}}_{n}\overset{.}{\alpha}m_{sn}\left\{ {{2z_{6n}} + {c_{1n}\cos \quad \eta_{n}} - {b_{2n}{\sin \left( {\gamma_{n} + \eta_{n}} \right)}}} \right\}} +} \\{\quad {{{\overset{.}{\eta}}_{n}\overset{.}{\beta}m_{sn}a_{1n}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} + {\overset{.}{\alpha}\overset{.}{\beta}a_{1n}m_{sn}{\sin \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}} -}} \\{\quad {{{gm}_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\cos \quad \beta} - {k_{sn}\left( {z_{6n} - l_{sn}} \right)} + {{{\overset{.}{z}}_{0}\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)}m_{sn}{\sin\left( {\alpha + \gamma_{n} +} \right.}}}} \\{\quad {{\overset{.}{{\left. \eta_{n} \right)\cos \quad \beta} - \beta}{\overset{.}{z}}_{0}m_{sn}{\cos \left( {\alpha + \gamma_{n} + \eta_{n}} \right)}\sin \quad \beta}\rangle}} \\{= \quad {{{- c_{sn}}{\overset{.}{z}}_{6n}} - {\lambda_{1n}\sin \quad \eta_{n}} - {\lambda_{2n}\cos \quad \eta_{n}}}}\end{matrix} & (112) \\\begin{matrix}{\quad {m_{sn}\left\{ {{\overset{¨}{z}}_{6n} + {\overset{¨}{\alpha}E_{2}} - {\overset{¨}{\beta}a_{1n}C_{{\alpha\gamma\eta}\quad n}} - {{\overset{.}{\eta}}_{n}^{2}z_{6n}} - {{\overset{.}{\alpha}}^{2}\left( {z_{6n} + E_{1}} \right)} - {{\overset{.}{\beta}}^{2}B_{1}C_{{\alpha\gamma\eta}\quad n}} -} \right.}} \\{{\quad \left. {{2{\overset{.}{\eta}}_{n}\overset{.}{\alpha}z_{6n}} + {{gC}_{{\alpha\gamma\eta}\quad n}C_{\beta}}} \right\}} + {k_{sn}\left( {z_{6n} - l_{sn}} \right)}} \\{= \quad {{{- c_{sn}}{\overset{.}{z}}_{6n}} - {\lambda_{1n}S_{\eta \quad n}} - {\lambda_{2n}C_{\eta \quad n}}}}\end{matrix} & (113) \\{\lambda_{2n} = \frac{\begin{matrix}{\therefore{m_{sn}\left\{ {{\overset{¨}{z}}_{6n} + {\overset{¨}{\alpha}E_{2}} - {\overset{¨}{\beta}a_{1n}C_{{\alpha\gamma\eta}\quad n}} - {{\overset{.}{\eta}}_{n}^{2}z_{6n}} - {{\overset{.}{\alpha}}^{2}\left( {z_{6n} + E_{1}} \right)} - {{\overset{.}{\beta}}^{2}B_{1}C_{{\alpha\gamma\eta}\quad n}} -} \right.}} \\{\left. {{2{\overset{.}{\eta}}_{n}\overset{.}{\alpha}z_{6n}} + {{gC}_{{\alpha\gamma\eta}\quad n}C_{\beta}}} \right\} + {k_{sn}\left( {z_{6n} - l_{sn}} \right)} + {c_{sn}{\overset{.}{z}}_{6n}} + {\lambda_{1n}S_{\eta \quad n}}}\end{matrix}}{- C_{\eta_{n}}}} & (114) \\\begin{matrix}{{{{\frac{\quad}{t}\left( \frac{\partial L}{\partial{\overset{.}{z}}_{12n}} \right)} - \frac{\partial L}{\partial z_{12n}}} = {{\frac{\partial F}{\partial{\overset{.}{z}}_{12n}} + {\sum\limits_{l,n}{\lambda_{1n}a_{1{n6}}\quad l}}} = 1}},2,{{3\quad n} = i},{ii},{iii},{iv}} \\{{k_{wn}\left( {z_{12n} - l_{wn}} \right)} = {{{- c_{wn}}{\overset{.}{z}}_{12n}} + {\lambda_{3n}\cos \quad \alpha \quad \cos \quad \beta}}} \\{= {{{- c_{wn}}{\overset{.}{z}}_{12n}} + {\lambda_{3n}C_{\alpha}C_{\beta}}}}\end{matrix} & (115) \\{{\therefore\lambda_{3n}} = \frac{{c_{wn}{\overset{.}{z}}_{12n}} + {k_{wn}\left( {z_{12n} - l_{wn}} \right)}}{C_{\alpha}}} & (116)\end{matrix}$

From the differentiated constraints it follows that: $\begin{matrix}{{{{{\overset{¨}{\theta}}_{n}e_{2n}S_{\theta \quad n}} + {{\overset{.}{\theta}}_{n}^{2}e_{2n}C_{\theta \quad n}} - {{\overset{¨}{z}}_{6n}S_{\eta \quad n}} - {{\overset{.}{z}}_{6n}{\overset{.}{\eta}}_{n}C_{\eta \quad n}} - {{{\overset{¨}{\eta}}_{n}\left( {z_{6n} - d_{1n}} \right)}C_{\eta \quad n}} - {{\overset{.}{\eta}}_{n}{\overset{.}{z}}_{6n}C_{\eta \quad n}} + {{{\overset{.}{\eta}}_{n}^{2}\left( {z_{6n} - d_{1n}} \right)}S_{{\eta \quad n}\quad}}} = 0}{{{{\overset{¨}{\theta}}_{n}e_{2n}C_{\theta \quad n}} - {{\overset{.}{\theta}}_{n}^{2}e_{2n}S_{\theta \quad n}} - {{\overset{¨}{z}}_{6n}C_{\eta \quad n}} + {{\overset{.}{z}}_{6n}{\overset{.}{\eta}}_{n}S_{\eta \quad n}} + {{{\overset{¨}{\eta}}_{n}\left( {z_{6n} - d_{1n}} \right)}S_{\eta \quad n}} + {{\overset{.}{\eta}}_{n}{\overset{.}{z}}_{6n}S_{\eta \quad n}} + {{{\overset{.}{\eta}}_{n}^{2}\left( {z_{6n} - d_{1n}} \right)}C_{{\eta \quad n}\quad}}} = {0\therefore\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}}} & (117) \\{{\overset{¨}{\eta}}_{n} = \frac{\begin{matrix}{{{\overset{¨}{\theta}}_{n}e_{2n}S_{\theta \quad n}} + {{\overset{.}{\theta}}_{n}^{2}e_{2n}C_{\theta \quad n}} - {{\overset{¨}{z}}_{6n}S_{\eta \quad n}} - {2\quad {\overset{.}{\eta}}_{n}{\overset{.}{z}}_{6\quad n}C_{\eta \quad n}} + {{{\overset{.}{\eta}}_{n}^{2}\left( {z_{6n} - d_{1n}} \right)}S_{\eta \quad n}}}\end{matrix}}{\left( {z_{6n} - d_{1n}} \right)C_{\eta \quad n}}} & (118) \\{{\overset{¨}{z}}_{6n} = \frac{\begin{matrix}\begin{matrix}{{{\overset{¨}{\theta}}_{n}e_{2n}C_{\theta \quad n}} - {{\overset{.}{\theta}}_{n}^{2}e_{2n}S_{\theta \quad n}} + {{{\overset{¨}{\eta}}_{n}\left( {z_{6n} - d_{1n}} \right)}S_{\eta \quad n}} +} \\{{2{\overset{.}{\eta}}_{n}{\overset{.}{z}}_{6n}S_{\eta \quad n}} + {{{\overset{.}{\eta}}_{n}^{2}\left( {z_{6n} - d_{1n}} \right)}C_{\eta \quad n}}}\end{matrix}\end{matrix}}{C_{\eta \quad n}}} & (119)\end{matrix}$

and $\begin{matrix}{{\overset{.}{z}}_{12n} = \frac{\begin{matrix}{{\left\{ {{\overset{.}{\alpha}\quad z_{12n}S_{\alpha}} - {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{3n}C_{{\alpha\gamma\theta}\quad n}} + {\overset{.}{\alpha}c_{2n}S_{{\alpha\gamma}\quad n}} - {\overset{.}{\alpha}b_{2n}C_{\alpha}}} \right\} C_{\beta}} - {\overset{.}{z}}_{0} +} \\{{\overset{.}{\beta}\left\lbrack {{\left\{ {{z_{12n}C_{\alpha}} + {e_{3n}S_{{\alpha\gamma}\quad \theta \quad n}} + {c_{2n}C_{{\alpha\gamma}\quad n}} + {b_{2n}S_{\alpha}}} \right\} S_{\beta}} + {a_{1n}C_{\beta}}} \right\rbrack} + {{\overset{.}{R}}_{n}(t)}}\end{matrix}}{C_{\alpha}C_{\beta}}} & (120)\end{matrix}$

Supplemental differentiation of equation (116) for the later entropyproduction calculation yields:

k_(wn){dot over (z)}_(12n)=−c_(wn{umlaut over (z)}) _(12n)+{dot over(λ)}_(3n)C_(α)C_(β)−{dot over (α)}λ_(3n)S_(α)C_(β)−{dot over(β)}λ_(3n)C_(α)S_(β)  (121)

therefore $\begin{matrix}{{\overset{¨}{z}}_{12n} = \frac{{{\overset{.}{\lambda}}_{3n}C_{\alpha}C_{\beta}} - {\overset{.}{\alpha}\lambda_{3n}S_{\alpha}C_{\beta}} - {\overset{.}{\beta}\lambda_{3n}C_{\alpha}S_{\beta}} - {k_{wn}{\overset{.}{z}}_{12n}}}{c_{wn}}} & (122)\end{matrix}$

or from the third equation of constraint: $\begin{matrix}{{{\overset{..}{z}}_{0} + {\left\{ {{{\overset{..}{z}}_{12n}\cos \quad \alpha} - {{\overset{.}{z}}_{12n}\overset{.}{\alpha}\cos \quad \alpha} - {\overset{..}{\alpha}z_{12\quad n}\sin \quad \alpha} - {\overset{.}{\alpha}{\overset{.}{z}}_{12\quad n}\sin \quad \alpha} - {{\overset{.}{\alpha}}^{2}z_{12n}\cos \quad \alpha} + {\left( {\overset{..}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{3n}\quad \cos \quad \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}\quad - {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)^{2}e_{3n}\quad \sin \quad \left( {\alpha + \gamma_{n} + \theta_{n}} \right)} - {\overset{..}{\alpha}c_{2n}\sin \quad \left( {\alpha + \gamma_{n}} \right)} - {{\overset{.}{\alpha}}^{2}c_{2n}\cos \quad \left( {\alpha + \gamma_{n}} \right)} + {\overset{..}{\alpha}b_{2n}\cos \quad \alpha} - {{\overset{.}{\alpha}}^{2}b_{2n}\sin \quad \alpha}} \right\} \cos \quad \beta} - {\overset{.}{\beta}\left\{ {{{\overset{.}{z}}_{12n}\cos \quad \alpha} - {\overset{.}{\alpha}z_{12n}\sin \quad \alpha} + {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{3n}\cos \quad \left( {\alpha + \gamma_{n} + \theta_{n}} \right)} - {\overset{.}{\alpha}c_{2n}\sin \quad \left( {\alpha + \gamma_{n}} \right)} + {\overset{.}{\alpha}b_{2n}\cos \quad \alpha}}\quad \right\} \sin \quad \beta} - {\overset{..}{\beta}\left\lbrack {{\left\{ {{z_{12n}\cos \quad \alpha} + {e_{3n}\sin \quad \left( {\alpha + \gamma_{n} + \theta_{n}} \right)} + {e_{2n}\cos \quad \left( {\alpha + \gamma_{n}} \right)} + {b_{2n}\sin \quad \alpha}} \right\} \sin \quad \beta}\quad + {a_{1n}\cos \quad \beta}} \right\rbrack} - {\overset{.}{\beta}\left\lbrack {{\left\{ {{z_{12n}\cos \quad \alpha} - {\overset{.}{\alpha}z_{12n}\sin \quad \alpha} + {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{3n}\cos \quad \left( {\alpha + \gamma_{n} + \theta_{n}} \right)} - {\left( {\overset{.}{\alpha} + {\overset{.}{\gamma}}_{n}} \right)c_{{2n}\quad}\sin \quad \left( {\alpha + \gamma_{n}} \right)} + {\overset{.}{\alpha}b_{2n}\cos \quad \alpha}} \right\} \sin \quad \beta} + {\overset{.}{\beta}\left\{ {{z_{12n}\cos \quad \alpha} + {e_{3n}\sin \quad \left( {\alpha + \gamma_{n} + \theta_{n}} \right)} + {c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} \cos \quad \beta} - {\overset{.}{\beta}a_{1n}\sin \quad \beta}} \right\rbrack} - {{\overset{..}{R}}_{n}(t)}} = 0} & (123) \\{{\overset{¨}{z}}_{12n} = \frac{\begin{matrix}{{\overset{¨}{z}}_{0} + \left\{ {{{- {\overset{.}{z}}_{12n}}\overset{.}{\alpha}\cos \quad \alpha} - {\overset{¨}{\alpha}z_{12n}\sin \quad \alpha} - {\overset{.}{\alpha}{\overset{.}{z}}_{12n}\sin \quad \alpha} -} \right.} \\{{{\overset{.}{\alpha}}^{2}z_{12n}\cos \quad \alpha} + {\left( {\overset{¨}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{3n}{\cos \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} -} \\{{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)^{2}e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} - {\overset{¨}{\alpha}c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} -} \\{{\left. {{{\overset{.}{\alpha}}^{2}c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {\overset{¨}{\alpha}b_{2n}\cos \quad \alpha} - {{\overset{.}{\alpha}}^{2}b_{2n}\sin \quad \alpha}} \right\} \cos \quad \beta} -} \\{{\overset{.}{\overset{.}{\beta}\left\{ {{{\overset{.}{z}}_{12n}\cos \quad \alpha} - \alpha} \right.}z_{12n}\sin \quad \alpha} + {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{3n}\cos\left( {\alpha +} \right.}} \\{{\left. {{\overset{.}{\left. {\gamma_{n} + \theta_{n}} \right) - \alpha}c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {\overset{.}{\alpha}b_{2n}\cos \quad \alpha}} \right\} \sin \quad \beta} -} \\{\overset{¨}{\beta}\left\lbrack \left\{ {{z_{12n}\cos \quad \alpha} + {e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} +} \right. \right.} \\{\left. {{\left. {{c_{2n}{\cos \left( {\alpha + \gamma_{n}} \right)}} + {b_{2n}\sin \quad \alpha}} \right\} \sin \quad \beta} + {a_{1n}\cos \quad \beta}} \right\rbrack -} \\{{\overset{.}{\overset{.}{\beta}\left\lbrack \left\{ {{{\overset{.}{z}}_{12n}\cos \quad \alpha} - \alpha} \right. \right.}z_{12n}\sin \quad \alpha} + {\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{3n}\cos\left( {\alpha + \gamma_{n}} \right.}} \\{\left. {+ \theta_{n}} \right) - {\left( {\overset{.}{\alpha} + {\overset{.}{\gamma}}_{n}} \right)c_{2n}{\sin \left( {\alpha + \gamma_{n}} \right)}} + {\left. {\overset{.}{\alpha}b_{2n}\cos \quad \alpha} \right\}\sin \quad \beta} +} \\{\overset{.}{\beta}\left\{ {{z_{12n}\cos \quad \alpha} + {e_{3n}{\sin \left( {\alpha + \gamma_{n} + \theta_{n}} \right)}} + {c_{2n}\cos\left( {\alpha +} \right.}} \right.} \\{\left. {{\left. {\left. \gamma_{n} \right) + {b_{2n}\sin \quad \alpha}} \right\} \cos \quad \beta} - {\overset{.}{\beta}a_{1n}\sin \quad \beta}} \right\rbrack - {{\overset{¨}{R}}_{n}(t)}}\end{matrix}}{\left( {{- \cos}\quad {\alpha cos}\quad \beta} \right)}} & (124)\end{matrix}$

IV. Equations for Entropy Production

Minimum entropy production (for use in the fitness function of thegenetic algorithm) is expressed as: $\begin{matrix}{\frac{d_{\beta}S}{dt} = \frac{\begin{matrix}{{- 2}{\overset{.}{\beta}}^{2}\left\lbrack {{\overset{.}{\alpha}m_{b}A_{1}A_{2}} + {m_{sn}B_{1}\left\{ {{{\overset{.}{z}}_{6n}C_{{\alpha\gamma\eta}\quad n}} -} \right.}} \right.} \\{{\left( {\overset{.}{\alpha} + {\overset{.}{\eta}}_{n}} \right)z_{6n}S_{{\alpha\gamma\eta}\quad n}} - \left. {\overset{.}{\alpha}\quad A_{4}} \right\} + {m_{an}B_{2}\left\{ {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{1n}C_{{\alpha\gamma\theta}\quad n}} -} \right.}} \\{\left. {\overset{.}{\alpha}\quad A_{6}} \right\} + {m_{wn}B_{3}\left\{ {{\left( {\overset{.}{\alpha} + {\overset{.}{\theta}}_{n}} \right)e_{3n}S_{{\alpha\gamma\theta}\quad n}} - {\overset{.}{\alpha}\quad A_{6}}} \right\}} -} \\\left. {{{\overset{.}{z}}_{0}\left( {m_{ba} + m_{sawan}} \right)}{S_{\beta}/2}} \right\rbrack\end{matrix}}{m_{saw2n} + m_{bal} + {m_{b}A_{1}^{2}} + {m_{sn}B_{1}^{2}} + {m_{an}B_{2}^{2}} + {m_{wn}B_{3}^{2}}}} & (125) \\{\frac{d_{\alpha}S}{dt} = \frac{{- 2}{\overset{.}{\alpha}}^{2}\left\{ {{m_{sn}\overset{.}{\alpha}{{\overset{.}{z}}_{6n}\left( {z_{6n} + E_{1n}} \right)}} + {m_{sn}z_{6n}{\overset{.}{\eta}}_{n}E_{2n}} + {{\overset{.}{\theta}}_{n}m_{aw1n}H_{2n}}} \right\}}{m_{b\quad b\quad l} + m_{saw1n} + {m_{sn}{z_{6n}\left( {z_{6n} + {2E_{1n}}} \right)}} - {2m_{aw1n}H_{1n}}}} & (126) \\{\frac{d_{\eta_{n}}S}{dt} = {{{\overset{.}{\eta}}_{n}^{3}{tg}\quad \eta_{n}} - \frac{2{\overset{.}{\eta}}_{n}{\overset{.}{z}}_{6n}}{z_{6n} - d_{1n}}}} & (127) \\{\frac{d_{z_{6n}}S}{dt} = {2{\overset{.}{\eta}}_{n}{\overset{.}{z}}_{6n}^{2}{tg}\quad \eta_{n}}} & (128) \\{\frac{d_{z_{12n}}S}{dt} = {{\overset{.}{z}}_{12n}^{2}\left( {\overset{.}{\alpha} + {\overset{.}{\alpha}{tg}\quad \alpha} + {2\overset{.}{\beta}{tg}\quad \beta}} \right)}} & (129)\end{matrix}$

The learning module 101 gains pseudo-sensor signals based on the kineticmodels of the vehicle and suspensions obtained by the above-describedmethods. Then, the learning module 101 directs the learning control unitto operate based on the pseudo-sensor signals. Further, at the optimizedpart, the learning module 101 calculates the time differential of theentropy from the learning control unit and time differential of theentropy inside the controlled process. In this embodiment, the entropyinside the controlled processes is obtained from the kinetic models asdescribed above. This embodiment utilizes the time differential of theentropy dS_(cs)/dt (where S_(cs) is S_(c) for the suspension) relativeto the vehicle body and dS_(s)/dt to which time differential of theentropy dS_(ss)/dt (where the subscript ss refers to the suspension)relative to the suspension is added. Further, this embodiment employsthe damper coefficient control type shock absorber. Since the learningcontrol unit (control unit of the actual control module 101) controlsthe throttle amount of the oil passage in the shock absorbers, the speedelement is not included in the output of the learning control unit.Therefore, the entropy of the learning control unit is reduced, andtends toward zero.

The optimized part defines the performance function as a differencebetween the time differential of the entropy from the learning controlunit and time differential of the entropy inside the controlled process.The optimized part genetically evolves teaching signals (input/outputvalues of the fuzzy neural network) in the learning control unit withthe genetic algorithm so that the above difference (i.e., timedifferential of the entropy for the inside of the controlled process inthis embodiment) becomes small. The learning control unit is optimizedbased on the learning of the teaching signals.

Then, the parameters (fuzzy rule based in the fuzzy reasoning in thisembodiment) for the control unit at the actual control module 101 aredetermined based on the optimized learning control unit. Thereby, theoptimal regulation of the suspensions with nonlinear characteristic canbe allowed.

Although the foregoing has been a description and illustration ofspecific embodiments of the invention, various modifications and changescan be made thereto by persons skilled in the art, without departingfrom the scope and spirit of the invention as defined by the followingclaims.

What is claimed is:
 1. An optimization control method for a shockabsorber comprising the steps of: obtaining a difference between a timedifferential of entropy inside said shock absorber and a timedifferential of entropy given to said shock absorber from a control unitthat controls said shock absorber; and optimizing at least one controlparameter of said control unit by using a genetic algorithm, saidgenetic algorithm using said difference as a performance function. 2.The optimization control method of claim 1, wherein said timedifferential of said step of optimizing reduces an entropy given to saidshock absorber from said control unit.
 3. The optimization controlmethod of claim 1, wherein said control unit comprises a fuzzy neuralnetwork, and wherein a value of a coupling coefficient for a fuzzy ruleis optimized by using said genetic algorithm.
 4. The optimizationcontrol method of claim 1, wherein said control unit comprises alearning control module and an actual control module, said methodfurther including the steps of optimizing a control parameter based onsaid genetic algorithm by using said performance function, determiningsaid control parameter of said actual control module based on saidcontrol parameter and controlling said shock absorber using said actualcontrol module.
 5. The optimization control method of claim 4, whereinoptimization of said learning control unit is performed using asimulation model, said simulation model based on a kinetic model of avehicle suspension system.
 6. The optimization control method of claim4, wherein said shock absorber is arranged to alter a damping force byaltering a cross-sectional area of an oil passage, and said control unitcontrols a throttle valve to thereby adjust said cross-sectional area ofsaid oil passage.